To put your money to best use, rest of your system specs would benefit us greatly
For one, we don't know if your motherboard uses DDR2 or DDR3 RAM, so if we were to suggest a CPU/Mobo upgrade, we may need to include a new pair of RAM.
Yeah...and unless we know what monitor resolution you have, we don't know where the bottleneck is happening.
For example, if you're playing at around 1280x1024, 1280x720 or 1600x900 (or less), I'd suggest upgrading the CPU.
Why? Because i have a GTX 560 too, and a core 2 quad Q8400. Now i play at on CRT monitor with 1024x768 resolution. It refreshes at 85Hz so vsync for me is 85 fps. However, in a lot of recent games (heck, even CoD MW2) the fps jumps around between 50 to 85 with adaptive vsync. so sometimes things get a bit choppy. I'd be fine with an LCD monitor and a 60 fps refresh rate, however.
BF3 MP on the other hand KILLS my CPU while the campaign is mostly fine. So i have a CPU bottleneck here (and in most other places where the fps jumps around).