I can't really explain my graphics problem in words. I made a recording using fraps. My graphics card is an ATI Radeon HD 4350. I used Skyrim in the recording, but it happens with almost all games. I have a Windows 7 (x64) computer with 3 GB RAM and an Intel Core 2 Duo processor.
Only had an ATI card once (~10 years ago) and showed the same symptoms. Sometimes new driver versions might fix it (if introduced by new drivers), but it's usually some hardware issue: its possible there's faulty memory on the card or the card is overheating. If it's not changed by different driver versions, it's usually something with the card, so if you've got another one to test, that would be the easiest solution.
Re'install card into another PCI-e slot, preferably a lower one, then retry. This happened to my card and i can only use it in PCIx4 and not x16, which is a bummer. Or completely remove all catalyst drivers then do a re-install. Worked for me a couple of times.