Hi, I determened what caused the problems. Here is the story.
I bought this new Gigabyte MBo and i7 CPU cause my old MBo (ASRock X58 Extreme) burned out in smoke literaly, what happened near the CPU (chip with 1R1 label). I knew that CPU (i7 920) was also destroyed cause the meltdown on MBo was so severe that it started to burn trough the back side of PC case. So I bought new MBo (GA-X58-USB3) and CPU (i7 950) cause I found them at bargain price of 350$ for both. I put the old and new components together and PC failed to post any beeps and screen was blank. Then I posted the initial question.
Today I went to my friends home and tested out my components separatly (new MBO and CPU, old GPU and RAM) on his comp.
I did not suspect further results.
It turned out that my old GPU (XFX Radeon HD5830) was also destroyed in the ASRock X58 Extreme meltdown, along side 1 of 3 sticks of my Corsair Dominator 1600 DDR3.
2 sticks (4GB) of DDR3 survived the meltdown, and probably work (only the 3rd stick reported error on boot when i tested them 1 by 1).
After finding all this, my conclusion is that a bad quality MBo is worst thing your PC build.
Although my ASRock X58 Extreme lasted 4 years without any problems, in one moment it wiped out itself, CPU, GPU and 1/3 RAM. Or maybe my PSU (OCZ Silent XStream 2)caused the meltdown?
Can somebody advise me is it safe to try to use my old PSU that was present during the meltdown, or is it probably faulty as well?