The issue with an eyefinity set-up should not be your CPU it will be your GPU. The 6790 is a decent mid-range card based on the same GPU as the 68xx series, I've run an eyefinity set-up using a 6870 and had good results.
It depends on what sort of games you will be playing and at what settings you expect. A 6970 @ 5760x1080 (~6016x1080 with bezel) was enough to run Crysis in eyefinity mode at low-medium, and games like BC3 and JC2 on higher settings but games like GTAIV and Mirrors Edge (hacked) will need minimum 2GB VRAM so dont expect them to run at all across three screens.
Your CPU should not be your bottleneck (depending on what your using it for), for games the limiting factor will be your GPU.
Ram doesn't add up when you crossfire. So you will have 2GB total shared between 3 monitors. So I suggest at least 2GB if you want to game with 3 monitors. That is a great 6950 card and you will be able to handle game a lot better.