I like the build and all, but personally I'd go with a z77 chipset instead oh h77. Even though you don't need it for this particular build, I think it's better to have options and not use them then to not have them at all. The big question about this build, though, is what resolution do you plan on gaming at?
I have a 6870, and 1366x768 is my perfect resolution. 1080p is doable for a lot of games, but hardcore games like crysis and BF3 are too much for it unless I lower the details a bunch. At 1366x768, I can turn the details way up and it plays great (I like high fps). However, if I were you, I'd get a 1080p monitor and play at a lower resolution or with lowered details. You'll thank yourself later on. Native resolution is always best, but it's not that big of a deal to me.
CPU: Intel Core i3-2120 3.3GHz Dual-Core Processor ($119.00 @ Mac Connection)
Motherboard: ASRock H77M Micro ATX LGA1155 Motherboard ($69.99 @ Newegg)
Memory: G.Skill Ares Series 8GB (2 x 4GB) DDR3-1600 Memory ($44.99 @ Newegg)
Storage: Seagate Barracuda 500GB 3.5" 7200RPM Internal Hard Drive ($64.99 @ Microcenter)
Video Card: Sapphire Radeon HD 6870 1GB Video Card ($149.99 @ Newegg)
Case: Cooler Master Elite 430 ATX Mid Tower Case ($39.99 @ SuperBiiz)
Power Supply: SeaSonic 520W ATX12V / EPS12V Power Supply ($59.50 @ Newegg)
Optical Drive: Lite-On iHAS124-04 DVD/CD Writer ($16.99 @ Newegg)
(Prices include shipping and discounts when available.)
(Generated by PCPartPicker 2012-08-10 23:49 EDT-0400)
I really wish I could fork over more money for a better gpu but i cant atm.
I will get a 1080p monitor.
Your HD6870 will be able to perform very well at 1080p anyway, I remember getting 40-50FPS @1080p (high preset) on BF3 Multiplayer with my X4 955 set up. Probably even more if you're just going to play Campaign on BF3.
You won't have any problems with MW3, however, since that game isn't TOO demanding.