Your going to be fine for Running D3, a 5770 should handle the graphics just fine, and your dual core should as well.
The game is being made primarily to enhance a storyline, not to be super graphics intensive or performance hindering.
If you dont have a normal amount of ram, lets say 4gb's of ddr3 then id say look into upgrading for that.
At this point all of this is "guess work" though as they haven't officially released system requirements yet. But judging by Starcraft 2's graphics engine, its safe to say a computer with a DX9 card or above, at least a dual core, 4gb of ram, and like 10-20gb of devoted HD space should run the game just fine