I've got the same problem with my z87-d3hp and a Sapphire HD5850 Xtreme.
Updating the bios to the latest version (F5c) and re-flashing my video card's BIOS didn't help at all.
Also, full HD is working perfectly with the integrated graphics of the 4770k (same DVI cable, same monitor)
Can we please get a BIOS upgrade to resolve this issue?