If you're not overclocking, you don't need an aftermarket CPU cooler. You do need a CPU cooler, but the one that comes with the CPU is fine.
You don't really need dual GPUs for the only game you've listed. Hell, you don't even need a single great GPU for it. I wouldn't advise starting a build with dual cards, as you spend a lot more than you should, you eliminate an upgrade path, and you really don't need that much power immediately.
EDIT: Revised build:
CPU: i5-2500 $210
GPU: HD 6950 $280 after rebate. Don't forget to check out the guides out there on how you can turn this into a 6970 for free with a BIOS flash.
Total: $790 with Corsair's sticks or $815 with G.Skill's.
I also forgot to ask if that $1,200 was only for the parts you've listed or the entire build (add a case, PSU, OS and optical). If you need the other parts, check out:
Case: HAF 922 $100
PSU: XFX 750W $100 after rebate
Optical: The absolute cheapest SATA DVD burner you can find. Usually under $20.
OS: Windows 7 Home 64-bit $100
Those would bring the total to $1,135.
If you don't need the parts you didn't list, up the CPU to the i7-2600, and up the GPU to the GTX 580. That would total $1,025.