Assuming you have connected the 2x6 pin PCIe power cables.
Also please mention the PSU used.
I've dealt with this Exact same problem , Exact Same Video card with a friends setup. I kinda faced the same issue myself(Also have the 6870), My issue was resolved by a re boot. Im assuming you have re booted many times. You mentioned your MB post fine, Do you see the windows loading screen in DOS ? Because my friend could see the BIOS and windows loading screen but when it was time to see the Desktop his HDTV would have no display. Did a driver wipe and re install. (Use Driver Sweeper)