hey Guys.
I'd like to cut in with the conversation. I have very similar config of rig as You Macjeje. I was getting black screens with my ASROCK ex4 and I thought it was my PCI-e slots. Had 2 cards 460 and 670. The last one I bought some time ago to wait for new cards (a month). It got burned so as the 460. Try not to use the 2nd PCI-slot. Looking through the posts it would seem that you got kinda high temps so 2nd PCI-e slot is out of the picture. I bought a new mobo which wasn't necessary. I also got myself a new card just to make sure it all works. I did all bunch of stuff to get this working.
I just list what I did. Now I have no problems with my rig but the symptoms were same as yours. Black screen even though you hear sound but after several times freezes and shut downs and cards damaged. I couldn't even see the post screen of the AS ROCK. I know the Bios was loading and I could hear system loading as well but nothing on the screen. On my 3770K I7 intel HD video everything was working no problems.
1st thing I would do try different cable with your PSU - GPU. maybe different slot in the PSU if your is modular.
2nd I would reseat the CPU since that may also cause the PCI0-e to act weird. Get it out clean the old thermal paste and get a new one and stick it right in the place again. (BTW I use my box cooler and temps never get over 50 gameplay and idle is around 34 and it got it set to 4Ghz.
3rd. DO NOT use the PCI-e bottom slot. I got my card fried that way since the crap airflow. Got all my fans disconnected since I was trying things to get rid of the issue I had.
4th try opening the case if it is closed to see if it helps.
5th try reinstalling system starting from scratch. (it didn't help with the issue for me)
6th if those things wont help. get everything out. clean it all up (I did that) put it all back in and try.
Since I have replaced my cards and I was forced to get everything out mean time cleaning it since I got a new mobo (I thought it was mobo) with a new card 780TI everything got back to normal. I went back to my ASROCK xt4 and had no problem even with my old system (imaged).
Try those and see what results you get.
Personally what I'd been through with this problem I'd start with pulling everything out. cleaning it and start again. Maybe try psu different cable or/and different PCI-e power socket mean time when you pull all the stuff out.
BTW. I thought it might have been the RAM. But I still use my Kingston and no problem. Maybe try using different sockets for your modules.
BUT THE BEST SOLUTION HERE IS PULLING EVERYTHING OUT AND CLEAN. - My opinion. Really