Like your rig mate. Got the 920 myself a gem of a cpu.
They are neck on neck. The 6970 is more to so call future proof with its 2gb of ram and paired in crossfire it outperforms the 580 sli at 5760/1280*1024. For 1080p the 570 is more than enough. It got less ram but how long do you intend to keep the gpu. More hungry for ram games we won't probably see after two gpus updates..
To put it ,ore simple. One monitor ->Gtx 570
If your planning on adding more monitors like 3x 1080p get the 6970 due to its better scaling at ultra high resolutions.
Hope this helps.
Do you have a decent psu?
+1 for the Sapphire toxic 6950. Bang for your buck at this performance point it wins hand down.
Otherwise (to the OP's original question) it's going to depend on what resolution your running, single monitor @ 1920x1080 get a 570, any plans to do multi-monitor or higher resolution then get a 6970. Also in terms of scalability if your looking to CF / SLI in the future the 6970 offers better value.
I would definitely get the GTX 570 as Nvidia's DX11 tessellation is superior to AMDs. When you look at the benchmarks online especially in games like Hawx 2 in DX11 with tessellation on the difference is substantial.
If you are running at a high resolution or using multiple monitors you can always get the 2.5 GB version of the card.