Hi there,
this is one of those issues that might take a little time to resolve, especially as it seems to be intermittent!
My first guess would be to check that the Northbridge heatsink hasn't come loose. They do seem to be a bit wobbly (that's normal) but just double check that it is seated properly. It maybe worthwhile removing the NB heatsink and replacing the thermal paste as this can sometimes help.
Why am a suggesting this? Well as the problem seems to be video linked the Northbridge would be the logical place to start looking as the NB Chip handles the throughput of data for video.
Another thing to check is your BIOS. Is it the latest version F5? If not you can download from here:
http://www.gigabyte.com/products/product-page.aspx?pid=3424&dl=1#bios and then use QFlash to update rather than the @BIOS utility. Also, it is worth checking that your video drivers are up to date.
Also, are you using any software to monitor your system temps? If so, what? A utility such as HW Monitor can help if the problem is temperature related as it will show you if anything is running very hot. But, a note of caution, it isn't perfect and sometimes the sensors on the motherboard do not work as they should so it is best used as a guide rather than as gospel.
If none of theses steps resolve the problem, post back and we will look at other things that may be causing the issue. To put your mind at rest, I would be very surprised if the Memory controller has been damaged unless you have been running your RAM at 1600 MHz for any period of time. Even a faulty RAM Stick shouldn't impact on the memory controller.