From Diablo Wiki
Diablo III is being patterned after Diablo I, more than Diablo II. As such it's a horror-themed, gory, gruesome, frightening game. The developers hoped to make it creepy in places, while providing enough variety in theme, tone, and visuals to keep from growing monotonous.
Fans liked those words, but the game's initial visuals were widely-criticized for being "too colorful" and the debate expanded into the notorious Art Controversy. In response to fan complaints about insufficient gore and gothic styling, the developers added more grit and grime and blood to many of the dungeons.
There is no shortage of bloody scenes and gruesome imagery in Diablo 3. One fan collected a bunch of representative images and worked them into a collage, just to prove it.
A major source of gruesome imagery in Diablo 3 comes from the critical hit graphics. When monsters die to a hit that trips for critical hit bonus impact, the result is usually spectacular, with the monster exploding in a style appropriate for the type of damage that did it in. Fire critical hits create flaming, melting chunks, physical damage detonates the corpse leaving ribs and blood everywhere, etc.
This can be seen in various screenshots, and it's quite impressive to witness when actually playing. Fans at Blizzcon 2009 reported occasionally knocking the skeleton out of a monster; the skeleton which would be sent sliding across the ground for dozens of meters, leaving along streak of blood the whole way. No, really.
There are quite a few screenshots and other images that have similar material in them. Enough that there's "gore" tag in the Image Gallery.
- Images tagged for "gore" in the Image Gallery.
Use Gore Sparingly
Blizzard has made no secret about the fact that they enjoy gruesome imagery and lots of blood and explosions in Diablo 3. Cartoonish gore and violence, but still violence. They think the trick to keeping it acceptable is in how they present the material.
Jay Wilson spoke on that in an interview from Blizzcon, August 2009. 
One example of one of the biggest ways we edited ourselves is … we have this cool system where we can hit a dude so hard that his skeleton flies out. It was awesome, really cool-looking. And we added several skills that did that; every time you hit somebody, their skeletons fly out. But I have to say, it got a little boring after a while! It became a little excessive. It took away the coolness of it. And so there, we felt like our overuse of it actually de-emphasized it, and we didn’t like that. We were like, “No, if we want to push the skeleton out of somebody, we want it to be a big deal.” I want to really like see it, and I want it to be a special event. And that’s probably the main way we’ve toned ourselves down, is to go back and say, let’s not go so overboard, that there’s nothing cool about the violence.
Establishing that "Diablo I" Mood
Many long time fans of the Diablo series feel that Diablo 1 was much creepier than the sequel. Diablo 1 had much lower quality graphics and cinematics than Diablo 2, but it had a dark mood and theme that was not matched by the more action-heavy sequel. The D3 Team has accepted and acknowledged this, and they've repeatedly stated that their design vision for Diablo 3 is more akin to Diablo 1 than Diablo 2, in terms of the mood and vibe.
Leonard Boyarsky spoke on this in an interview with IGN in July 2008. 
Content Restrictions and Mature Rating
One problem with making the game as gruesome and bloody as previous titles in the series is that the game is marketed and sold internationally. Many nations, especially in Europe, have strict ratings guidelines about how much violence, blood, and gruesome content can be shown in a game. This forces lots of action and horror games to censor themselves, in order to be released in those regions.
Jay Wilson: Yeah, we’re going to have to be able to turn off blood, change the color and things like that, because you can’t have red blood in some regions, regions that we would very much like to sell the game in. So we definitely build everything, that every bit of gore, in a deposited manner so that at a future date, we can go through and change it all or turn it off. In terms of what kind options we give, we actually give within a particular version. We’ve haven’t nailed it down, but if you turn down the gore, you can actually change it to not have red blood. That seems to be really the sticking point for a lot of people because a lot of times we use blood as feedback. And so if we take that out, that actually hurts the gameplay. But we can change the note of that feedback so that it’s something that people are more okay with.
Wired.com: You’ll obviously have to edit content for regions like Germany and Australia, but what about China? Is that a more difficult case?Jay Wilson: Definitely for regions like Germany and Australia, we will have to change blood if we’re going to sell there. And that’s fine. Those are the standards for those regions, and we don’t really have a problem with catering to what they need and what they want. But China’s going to be hard for us. Because a lot of the restrictions there are really… we may not be able to do them. It may not be possible. With our relationship with NetEase, we recently got new information about what China really wants, and it’s a lengthy list. It’s really hard for us to cater to. We’ll try. There’s no reason we wouldn’t want to go there, but there is a certain point where we’d have to redo so much of the game that it’s not viable anymore.
Less Blood Options?
Early in development, Jay Wilson said they were planning options to turn down the gore and blood, and that the game would likely be censored for violence in some markets (such as Germany and China) that traditionally allow less violence in games.
The edits appear to not have been needed. Diablo III was approved without edits by the ratings boards in almost every region, and if there were any edits or gore removals required to gain rating approval, (without being tagged with an Adult level of rating that would have lessened the game's potential sales) Blizzard has not shared any details about it.
As of December 2011 there were no plans to include any sort of "less gore" player control option. 
This came as something of a surprise to many players, as many changes as the developers have included in Diablo III in order to make the game more accessible to the masses.
Pentagrams and Crosses Removed
Pentagrams, crosses, and other traditional Diablo-style holy symbols are not present in Diablo III, and have also been retconned out of World of Warcraft in recent expansions. This is an intentional design choice by the developers; in addition to helping them avoid potential controversy, they feel that removing symbols from existing real world religions will help them make their games more original and self-contained.
This change has created a lot of controversy, with fans objecting to the change on aesthetic, symbolic, and "gutless corporate weasel" grounds. Bashiok addressed this issue at length in October, 2010.
Bashiok: A lot of what we do now and have been doing for the past... 8 or so years, around the Warcraft III and initial World of Warcraft development era, is putting a lot of effort into crafting unique worlds. Meaning that we put more effort into every aspect of the game world itself. Who is this secret organization? What is their history? What’s their look? Do they have a specific accent? Do they have a symbol or mark that can identify them?
In the previous games I think there was a sense of “Oh it’s a demon, Diablo, devil, Satan, satanism… pentagram!” and while certainly there are those influences, what we’re creating now and have been creating for quite some time is evolving the game world and making something more complete. A complete world where every aspect is fully realized, maybe even if it doesn’t show up in the game.I think that same sort of world-crafting can be seen between Warcraft II and Warcraft III. A big shift in focus to fleshing out a unique fantasy world and story.
See the Pentagrams article for more on this issue, including a critical reaction from Max Schaefer, one of the creators of Diablo and an art director for Diablo I and Diablo II.
The first and longest appraisal of how well Diablo 3 kept to the gruesome theme established in the earlier games in the series came from Flux, a longtime fan of Diablo 1's horror vibe, who wrote extensively about the mood and gore in a report from Blizzcon 2008. 
- Blood, Guts, and an M-Rating
- One of my questions going into Blizzcon was about Diablo III’s tone and mood, and how much gruesome background art and blood would be present. Most players feel that Diablo I was much more of a horror game than Diablo II. Everyone who played the first title has a very clear memory of how scared they were when they first heard the Butcher give his “Ahh, Fresh Meat!” battle cry, and the dungeons, especially the Hell levels, were well-decorated by bloody, naked, dismembered bodies on stakes. Diablo II had plenty of blood and gore and horror elements too, (check out the tortured corpses and moats of blood in the Act 2 Sewers and Durance of Hate on your next run, and there are some nice gory wall decorations in dungeon as well), but it didn’t have the same sort of creepy, ominous, horror tone that D1 had.
- The D3 team has often said that they were going to recreate the mood and theme of D1 more than D2, and that they wanted to make D3 more of a horror game. I was dubious of this claim going into Blizzcon, but was pleasantly surprised by how horror-filled D3 is. The tone was set with gore and corpses in the very first scene. New players started off in a tiny encampment with just two NPCs. One of them was a talkative soldier, but the other was a silent meat wagon driver who spent his time endlessly shuffling between a wagon stacked high with bodies, from which he kept pulling corpses that he carried over and dumped onto a burning pyre. The animation was great, and the bodies were very well drawn as well. They looked like corpses, bloody and murdered ones.
- The mood continued as soon as a player moved out into the ruins of Tristram. The scenery was dark and oppressive, and no, the screenshots don’t at all do it justice. The floating, partially transparent mist looks so much better in the game than in the screenshots, where it just makes things look smudged and blurry. The black, gnarled trees, clusters of crows that flew away when the player got close enough to trigger them, dozens of ruined, blasted houses you could run through, bodies lying here and there, and small bunches of zombies to rout gave the area a great, creepy, doomed mood.
- Playing in the Blizzcon press room with crowds of journalists around, bright lights overhead, the only game sound through tinny headphones, and my attention mostly on gathering information, rather than enjoying myself, I didn’t feel any tension or atmosphere. I still enjoyed the visuals and theme, and found myself wishing I could play it at home, in a dark room with some candles burning, and the sound up high. It would be shivery. It will be shivery. I just hope Blizzard doesn’t have to tone down the gore and gruesome visuals too much, for the ratings boards.
- There were a number of nice set pieces elsewhere in Tristram that added to the mood. A human hand is seen at one point, clawing at the earth, before being yanked down into a dark cellar from which come horrible screams and a fountain of blood. Ghosts wander the streets, sobbing quietly and pathetically. Zombies are seen gnawing on corpses, and their moans and groans are very horror movie appropriate. The gruesome mood isn’t continued into the dungeon, but the design and graphics of the dungeon areas were very effective. They weren’t trying to be scary, but the monsters are so well animated and formed that they are threatening and very real. You want to destroy them, and even though the zombies you find early on are basically just walking experience pots, they’re loathsome and a little bit scary. I was again reminded of Diablo I, where even the weakest starting monsters were somewhat threatening, emotionally, if not from a survival standpoint.
- The death animations of the monsters are well done, and they’re very visceral and bloody. Normal deaths are quite messy, with body parts breaking off in gratuitous fashion and blood spilling, but the real fun comes from critical hit deaths. When monsters die from critical hit damage, (which happens quite often), they get an extra special death animation. They literally explode, in appropriate fashion for the type of damage that did them in. You’ll see frost on the shattered pieces, or lightning flickering along with an extra nova blast, or flames charring the chunks of meat, or gouts of blood from the corpse explosion like critical hits with physical damage. I was frequently impressed with how much blood spilled onto the floor during battles, and the ensanguinated stone and scattered arms, legs, heads, chunks of rib cage, and various internal organs left lying around after a big fight was just awesome. (In the game I mean. Not real life. Do not want FBI visit!)
- On the whole I was quite pleasantly surprised by the gruesome nature of D3. Much more than I’d expected, and more reminiscent of the mood of D1 than D2.
Screenshots seem to back up this appraisal, though we'll have to see more first hand descriptions of the game to know how well the D3 Team carries this approach through the entire game.
Various images showing off the gore and terror in Diablo III. Probably don't send the link of this part to your mother.