At that resolution it will be able to play just about every game on high, though you may run into poor performance if you try to max out the Anti Aliasing. Tessellation and Advanced Depth of Field may also slow things down. The most demanding games out there right now like Crysis 2 with DX11 or Metro 2033 will also probably give you trouble at that resolution.

At your resolution CPU performance is also quite important, as the GPU has less work to do at such a low resolution. For most games you probably will be fine but some of the really CPU heavy titles like Battlefield Bad Company 2 or Grand Theft Auto IV may run a bit slow on an original Phenom regardless of what GPU you have.
 
As Supernova said, a HD 5670 is a entry level card, and at 1024X768 resolution, I'd say it would provide good FPS and many modern, dual core games would run on high. But games like Crysis and Metro 2033 and GTA IV, BFBC2 ( all said by supernova ), would run a bit slow. But sice it's at that resolution, then it's okay. Also GTA IV is a heavily CPU dependent game, it requires a decent quad core and a good CPU to run it well.