Do you have a way to connect an old VGA cable to your GPU and monitor? (Not the DVI/HDMI). I had this problem recently that I got a black screen and not booting (100% convinced the board or GPU was shot), but what it was was that it (unbeknownst to me) sat at the BIOS screen, telling me some error, overclock error or CPU fan etc...but the display was only visible through the ol' VGA connection, NOT the DVI. (Sometimes the DARNEST things can happen).
Then it can also be with your PCIE slots, means take out the GPU, clear slots with air etc, possibly support the card with something (cardboard etc.) so it doesn't bend down too much. Here again, sometimes the must stupid things happen, not so much hardware error but more like mechanical. I'd guess the 290x would have the same problem that it's really heavy so I'd look into that too.
* If you can, boot with ONBOARD GPU activated but with the R290x in a slot, does it see the second GPU in device manager?
** At this point seriously consider that it MIGHT work but hanging at BIOS, unable to boot...so see whether you get picture through the VGA port. Also, you can look into BIOS settings with the onboard GPU, maybe you need to change something so you get a picture with the R290