I've experienced that problem before and there are two common things that could cause it. The first, and the easiest to remedy would be from the PSU. Which one are you using? The 6870's require two 6-pin connectors at the end after you plug it in. Did you remember to insert both of them? The min. recommended is about 500W.
The second problem could be your motherboard. On my previous computer, I had an older Asus motherboard with an NVidia chipset and it refused to play right with a new HD 6850 card I just bought. I tried everything... even updating the BIOS. Even the manufacturer sent me two different cards and neither of them worked. After further research I found out that some older motherboards with NVidia chips just refused to work with AMD cards... period; It worked flawlessly with my 8800GTS-512 that I previously had. My only solution to that problem was to just update my motherboard.