MSI GTX 560 3 monitors 3840x1024

The 5xx series requires 2 cards to push 3 monitors. Sorry bud.

Well, first of all, if you want to run two cards in SLI you need to make sure (1) your motherboard has support for SLI and (2) you have a decent and powerful enough PSU. I'd say at least 600W, preferably 650.
On all 3 screens put together the total resolution is gonna be about twice the resolution on a Full HD monitor screen. Two GTX 560 in SLI will be able to run BF3, although probably not on high settings, at least not with a smooth framerate.

As I see it you have 3 options:
1) If you don't mind playing on medium settings, you can try to find another 560, hopefully for cheap (if it's used it's worth not much more than 50 bucks I'd say, if it still works properly). This is by far the cheapest option.
2) If you wanna game on ultra settings, and use two cards, you're looking at least a 660 in SLI, but probably a 760 SLI. The website hardware.info had SLI tests and with the GTX 660 and on BF3 ultra on a Full HD monitor they got close to 90 fps. So on your resolution you're looking at 55 fps, however the testers do use a high end CPU which doesn't bottleneck the GPU so unless you have a fast i5 or i7 you'll probably get a little less fps, around 45-50. A 760 SLI is a safer bet, you'll probably get 60+ fps with BF3 on Ultra with 3840x1024 pixels. So 760's in SLI is probably the best bet, 2 new ones will probably cost you a little over 400 dollars.
3) If you wanna game on ultra, and use only 1 card, you'll need at least a GTX 780 period. Not lower than that. In that case I'd wait a couple days until the GTX 970 goes on sale, which is presumed to be a little faster than the 780. Anyway these cards can run 3 monitors, and can run BF3 on ultra with probably 50-60 fps on 3840x1024. A 780 sells for about 400 bucks, and the 970's price is not yet known but will probably be a little less than 400 dollars, so this is slightly cheaper than option #2.

If you wanna use AMD instead of Nvidia, then replace the Nvidia cards I mentioned with AMD ones (760 is equivalent to an R9 270(x); 780 is equivalent to an R9 290(x)) and replace "SLI" with "Crossfire", although your motherboard has to support Crossfire then.

Thats a pretty pricey machine for what you get, mind you it will do its job. If you can get something else I would look at that. The 81xx were turkeys slowerbclock for clock than phenom 2 they replaced, mind you they were clocked higher. Is 1200 your budget? Where do you live and what stores can you order from? Are there other options?

Yeah don't get the 8120 - get the FX-8350, it's a lot better. And that 280x will do fine probably, but that card will be pretty outdated soon. In that case I'd rather get a much faster GTX 970 which will come out in a few days and won't be too expensive (according to the rumours +/- 350 dollars). Also, unless you already have one, get a good CPU cooler like a 412S. You also won't need a 1000W PSU. For single 970 a 500 or 550W PSU will be enough (80-plus gold). If you might do SLI in the future, for 2-way SLI get a 700W supply. (For the 280x, get a 650W for single card, and 850W for 2-way Crossfire.)
That said, the 970 probably will cost you at least 100 bucks more than the 280x, but for gaming you'll have a much faster system, especially for future games.