I want to know the answer, too. Image quality is as important as frame rate, but now most review sites only do benchmarks. Right now most people are taken image quality as granted, and that could lead either company to reduce image quality in order to look good in benchmarks. And the hard part is that in this particular case, you have to see it in motion, screen shot won't do it.
For others who don't know, this is the site that discovered the shimmering in recent Radeon cards:
http://www.tweakpc.de/hardware/tests/grafikkarten/amd_radeon_hd_6870_hd_6850/s09.php
The card that produces the least shimmering is the previous high-end Cypress, but even that is not as good as GF100/104. So you have to wonder if the current high-end Cayman is at least as good as Cypress, or fall to the same category as Bart.
Unfortunately, even the site didn't do the same test for HD6970/6950. They only mentioned it brifely:
http://www.tweakpc.de/hardware/tests/grafikkarten/xfx_amd_radeon_hd_6970_hd_6950/s07
Base on translation, it seems they think it's originated in hardware, and AMD knew about the problem. But without a proper test, there is not way to be sure if it ever gets solved. In the meantime, those who wants the best image quality, at least in motion (which I think is more important in games), I have to say NVIDIA cards is the better bet.