I can't really answer your question, but I may be able to give you a pointer:
The AMD 780G chipset includes embedded AMD (nee ATI) graphics. You will therefore have the same issue with the graphics part of the chipset as you would if you had a discrete AMD/ATI graphics card of a similar specification and needing similar drivers.
The issue with all of these more capable graphics cards is that there are either no fully-capable open source drivers or the open source drivers are a little bit primitive compared to the closed source ones.
This gives the distros a bit of a tough decision; includes (as a default???) drivers that may be a bit rough around the edges, but are free? Include proprietary drivers that are not free (as in free), not debuggable and possibly not available in 64 bit form (if the supplier of the blob supplies a 64 bit blob, then fine, otherwise no)? Or just a 'lowest common denominator' driver that really doesn't take full advantage of the hardware and will not satisfy many people?
For any particular distro, there is probably a repo that has some extra, not-really-free-as-in-free software, that includes drivers for graphics chipsets. If you enable that, grab the drivers from the repo, you should have a good chance of progressing, but the procedure will vary from distro to distro. Which distro were you most interested in trying?
Install a SERVER version of 64 bit LINUX (Don't care which distro),
And, btw, the traditional (proper!) version of a server install wouldn't include any GUI, so this wouldn't be an issue....