Please let me know which processor you are using on this system?
There is a new BIOS version 0116 that you may want to try. You can download the file on the following link: https://downloadcenter.intel.com/SearchResult.aspx?lang=eng&keyword=Intel%20Desktop%20Board%20DH61be&sort=Date&sortDir=descending&perpage=10&DownloadType=BIOS
Please test the board out of the chassis using minimum configuration such as Motherboard, processor, 1 memory stick and power supply. No video card.
Make sure you have the latest graphics drivers installed https://downloadcenter.intel.com/SearchResult.aspx?lang=eng&ProductFamily=Graphics&ProductLine=Desktop+graphics+drivers&ProductProduct=Intel%C2%AE+Pentium%C2%AE+Processors+with+Intel%C2%AE+HD+Graphics&ProdId=3524&LineId=1100&FamilyId=39
Please test another PCI-e device on the board in order to discard it is defective.
The following link will show you the list of tested peripherals with a 6 series motherboard. There are some ATI video cards; however the one you are using is not there. I recommend you to contact ATI and ask them if they tested this video card on a 6 series board. http://www.intel.com/support/motherboards/desktop/dq67ep/sb/CS-032323.htm
Since the video card is working on the DG41RQ motherboard, you may want to try updating the firmware of the device. Then, test the video card again on the DH61ZE motherboard.
If you continue having the same behavior after all this procedure, it has to be a compatibility problem between the board and the video card.