See, this is a hard build because you want to buy a monitor, all your components, all your peripherals
and windows for < $1000.
In reality, this turns it into more of a $650-700 build. At the same time, you want to play BF3 on ultra settings. This requires quite a bit of horsepower, even with a single monitor.
So let's try to shave some money off of the basic build.
ASUS M5A97 - $99.99
AMD FX8120 Bulldozer - $169.99
Kingston HyperX 2x4GB 1600Mhz $48.99
Stock AMD heatsink - $0
XFX Radeon 7870 Double D - $299.99
Antec HCG-520M 520W Powersupply - $64.89
WD 1TB Black - $114.99
Bitfenix Shinobi - $69.99
Total: $868.83
I removed the heatsink (use the stock HSF, but don't overclock) and spec'd slightly cheaper RAM and powersupply. I also replaced the SSD with a 1Tb harddrive.
I'm trying to spec quality components that I know are going to last you a long time. Hence why I'm using a slightly more expensive ASUS board than the Gigabyte that you initially spec'd, same with the WD Harddive vs the Seagate that you spec'd. I know that these components/manufacturers are solid and should give you years of service.
You'll probably need to beg/borrow/steal a DVD drive from a friend in order to install Windows. But really, they aren't necessary after that in this day and age, which is why I didnt' spec one.