Another slight edit:
The official Diablo 3 Game Guide is now live, and includes an introduction which sets the stage for new players entering Diablo 3 without the background of the previous games. I believes backs up the updated content in my first edit.
Now, a star has fallen from the heavens—a grim omen that heralds the End of Days. Sanctuary’s frightened inhabitants are turning to abandoned legends and prophecies for answers. They have no hope against a full onslaught from the Burning Hells. The heroes of the past are gone or dead.
A new champion must rise…
So I've done some additional research, and I believe I have found the most credible Blizzard sourced information on the subject. At this link you can see a video in which a Blizzard panel member from Blizzcon 2011 explains that generally, you don't know what happened to the heroes from Diablo II. They specifically state that all that is known are hints and vague references, deliberately. They do go on to confirm a special case where you run into the apprentice of the Necromancer from Diablo II during a chance encounter in Act II. This thread on Blizznet also discussed some of the known lore related to the Necromancer's apprentice.
You come across him and his three skeleton minions, and he asks you to help him fight off some restless ghosts that the Cultists disturbed. In dialogue, he mentions something to the effect of his master being one of the heroes who helped defeat Diablo 20 years ago.
It is also confirmed in that same thread that the male Barbarian from Diablo III is not the same Barbarian from Diablo II, even though this is often speculated - since that was an original plan by Blizzard that was scrapped due to gender differences, and difficulty in plot integration.
Beyond those two knowns there is some speculation about the Sorceress from Diablo II being tied to a Blizzard short story that provides the backstory for the female Wizard but that is outside the game, and speculative.
There is also mention that by Brother Malachi, the healer in Act 1 that "Zakurum has fallen to disgrace," which bodes ill for the Paladin from Diablo II, if he is still alive.
From what I have researched, the Druid, Amazon, and Assassin are not mentioned at all.
So, I'll leave my original answer below with the grain of salt that there does not seem to be a solid reference. It is a widely distributed version of history at this point that Blizzard neither confirms nor denies, to my knowledge.
According to the wiki and numerous other sources Diablo 3 takes place 20 years after the events of Diablo 2, so the previous adventurers may be too old to swing morning stars against the minions of Hell at this point. The wiki also specifically mentions that Tyrial sent the previous heroes off to live in safety as a reward for their deeds in the last game. So my old Barb is likely bar tending at the Masculine Grunt Inn, telling patrons "This is yours... For you... A gift for you."
The game takes place in Sanctuary, the dark fantasy world of the Diablo series. Sanctuary was saved twenty years prior by unnamed heroes in Diablo II. Having survived that onslaught, Tyrael rewarded the heroes by sending them to safety. It is up to a new generation of heroes to face the forces of evil threatening the world.