Now I bought an i5 3570K and an ASRock Z77 EXTREME4. This costs exactly 500 dollars. The rest comes from my old system - an EVGA GTX 470 (1.28Gb, bought in 2010) and 6 Gb DDR 3.
I had to install Windows from a USB stick because I forgot to buy an optical unit (old one is IDE lol). Protip: if during USB installation Windows asks you to load a driver, cancel, go back to the welcome screen, pull out the stick, insert it in a different USB slot and restart the process. Don't try to use your old Windows installation with a new motherboard.
Crysis 2 max settings, with all the DX11 stuff on at 1920x1080, floats around 30fps. Stuttering is gone. Far Cry 3 also runs A LOT better (according to this the jump from a Q6600 to a 3570k doubles performance). Haven't tried much else but I'm happy. The GPU will be the first thing to replace but right now it's enough for everything.