So I was viewing the just announced Radeon cards, and I saw that 3DMark Firestrike score and I thought, maybe I need a new GPU or CPU. So I downloaded 3Dmark and tested it.
I'm not sure why you can't run Far Cry 3 on medium setting. I have a 6870 and I can run Far Cry 3 on high-ultra in 1080p just fine. Even though I'm running an i7 950, I don't think that there is much of CPU bottleneck in your case.
That is great but, I really think there is a problem in my PC. And I don't know how to search for it/find it. I've seen people play Crysis 3 on medium settings with the same parts and no problem, while I can only play on the lowest settings
Download the latest drivers for your GPU, check your CPU usage while playing (90%+ probably means that it is bottle-necking your GPU), and keep in mind that this could be a problem for certain games that rely heavily on the CPU. And also make sure to pay attention to your temperatures.
This I expect you already did but try turning down some of the graphics filters and Anti aliasing down.