None of the old characters will appear in Diablo III

The Diablo 3 Team has confirmed that the Barbarian will be the only recurring character class from Diablo II, of the original 5 Diablo III characters. More classes could be added in expansion(s) and some may be revisions of earlier character types. It`s also known that the Diablo III dev team at once though about making a Dervish class, but no information if that idea is completely shelved. It seems highly unlikely that the Dervish would appear though, as Blizzard have avoided talking about upcoming classes, and they would be unlikely to mention the class in passing if that is one class that will make the game. 

The Barbarian is the only Diablo II character to make it into Diablo III release. He's quite updated in style and use. 

The Witch Doctor is a new character, similar in use to the Necromancer. 

The Wizard is the third announced class, filling the archetypal magic user role previously occupied by the Sorceress.

The following characters have been confirmed to NOT appear: 

The Amazon will not appear in Diablo III. The Assassin will not appear in Diablo III. 
The Druid will not appear in Diablo III. 
The Necromancer will not appear in Diablo III. 
The Paladin will not appear in Diablo III. 
The Sorceress will not appear in Diablo III.

Some of these seem likely to join in on an upcoming expansion, that has been hinted about...

Jay Wilson said the following in an interview:

The Barbarian is the only returning class. Originally we actually planned to have no returning classes, but as we developed the barbarian, it essentially ended up turning into a barbarian. We sort of got to the point where we went – because we actually had a different name for him originally - I think were going to call it this other name and everyone else is just going to call it the barbarian, so maybe we should just call it the barbarian. 
One of the things we really did not want to do – it's why we set that initial goal to not repeat classes – was, we didn't just want a rehash; we wanted to make the sequel and the sequel has new gameplay and new experiences. I know a lot of people really love the classes in Diablo 2 and it's not my desire to intentionally arc those people – I like those classes to; I love the Necromancer, he's my favorite class from Diablo 2 – but as our goal we really wanted to provide new experiences; that was our focus.