AMD has 7870 which when overclocked trades blows with a stock 580 if not a bit faster. At stock it is about 5-10% faster than a 6950 in most benchies.
With these cards you will need to play at 1080p or lower, and without any AA most of the times.
Crysis and Crysis 2 would be even more demanding can't be maxed out at 1080p even without AA. I assume that you need avg FPS 60 and min around 40s. I know because I had a 6970 which is in the same tier as these cards.
My 7970 1175 1625 manages about 50+ FPS avg in Crysis 1 with min about mid 30s or so. Just about playable, decent, not perfect. On a stock 7970 it would be unplayable for me because the avg FPS would be 38-41 or so and min barely 30. This is at 1440p. A GTX 580 stock / 7870 OC would barely manage 1080p without AA without AF and avg FPS around 50s or so.
Crysis 2 1080p Ultra with High res pack with Dx11 I manage mostly 65-80+ with min in 30s and 40s. At 1440p it even goes as low as 20s and isn't playable. To play at 1440p I need to remove Dx11 and His Res pack to still get a respectable minimum of 50s or so.