AMD and ATI have a thing going that is somewhat unreliable.
err... AMD owns ATI as far as I'm aware. I only have experience with the older models of both companies products (ATI Rage pro, early Radeon cards and K6-2+ CPUs) so I have no idea how they are today.
What exactly do you mean, I guess, is what I'm asking. The CPU really has no bearing on the selection or use of graphics cards; but it does affect the choice of motherboard, which is the critical component when selecting a graphics card, as well. It's more or less the selection of motherboard that can make or break a good build (the bad tasting egg ruins the omelet...), however, I believe that boards for both Intel and AMD chips of course can both be found that are of good quality and generally reliable.
Basically- the question was wether to select a Intel, or AMD chip- BUT- selecting a different chip also means a different motherboard, which can change the feature-set and price considerably; I'd suggest exploring both options- configure a Intel build, and configure a AMD build within your price range, and then consider them based on a comparison of their features, such as the RAM speed of the motherboard and so forth.