Allrighty. It is painful to read about so many people having trouble with this board.
I have the same board and i7-3770 CPU, and GA 6870 graphics. And I also had problems with a clean install. It failed after Multibeast. Here is how I got it working.
First, I took out my 6870 graphics card. The HD4000 will do for now, and I may sell this card.
Make sure you read the guides I have linked, in advance, and that you understand each step.
I have read the BIOS settings guide for 7-series Gigabyte boards here:
http://www.tonymacx86.com/99-quick-guide-configuring-uefi-gigabyte-s-7-series-lga-1155-boards.html
I followed that guide exactly.
Then I have read the ML installation guide, and followed it exactly:
http://www.tonymacx86.com/61-unibeast-install-os-x-mountain-lion-any-supported-intel-based-pc.html
NOTE: You have to make your USB install drive before you start the whole process. During the OS install, if you are installing on a clean HD, it is better to have no other HD connected.
For the Multibeast 5.0.2 settings, I have used the settings in this post (#6 in the thread), EXCEPT the SSDT setting, which you don't need. I applied the Multibeast settings one at a time, and rebooted after each successful application:
http://www.tonymacx86.com/mountain-...your-multibeast-5-screenshots.html#post414740
Sound, LAN, and HD4000 all working very well for me so far.
At this point, you could put back your 6870 based graphics card.
If you did a clean install on a new HD, like me, the Mac App store might show that you have already installed your purchases, and you have no way of installing them again. Don't panic. Eject the drive volume with your old Lion install (Finder will let you, even if it's internal), and re-lauch the Mac App store. Now the store can't find your old installs, and will let you re-install.
Good luck!