I've never played a Diablo game but after watching the Diablo 3 playthroughs on YouTube, I must say I'm really excited about this game. But, do I need to play the older games to get any kind of satisfaction in terms of story, characters and gameplay.
Think Mass Effect. If you haven't played Mass Effect 1, you're not gonna understand the story of Mass Effect 2 all that much, and you probably won't understand any of the characters since many of them were l introduced in the first game. On the other hand, the combat gameplay is very different in ME2.
TL DR: See title.