I borrowed a HD6950 video card. Seems the problem is solved. Max rate seems to be 140hz, which is good enough. Tried 1280x720, and was able to run at 85hz, so it looks like the HD5000 issues were fixed, at least in this video card.
On a side note, the HD6950 was only about 10% faster than my current HD4870 at 1280x960 with no AA (using CRT monitor). GPU speed only increased from 750mhz to 800mhz, and although the memory speed went from 900mhz to 1250mhz, apparently the main bottleneck is the GPU.