It could potentially be a bad cable as well.
Okay, I've changed my PSU a couple days ago and I haven't had a single crash in DX9 yet. DX11 seems stable so far, too.
Thanks a lot everyone, especially Raist, for your help!
So I'm actually using a GTX980 as well right now and haven't encountered this issue. However, I'm not at all saying your issue isn't happening. What it sounds like to me, honestly, is that something is shorting out or perhaps isn't seated right in your PC. Could you take a look and make sure that your RAM and your new GPU are both seated firmly and properly into their slots? And there is some software I believe that can test for faulty hardware although I can't recall which I used in the past.
The reason I think it's an issue with something not being seated properly is because I encountered this exact issue with random restarts with no warning at all - just a blank screen, some sound, and BOOM - restart - for a long time back when I was primarily playing Rift and FFXIV wasn't out yet. The issue turned out to be that I had a bad stick of RAM and, what I think was happening, is that when my PC would go to read or write from the bad stick's memory, it would cause a restart.
FWIW I discovered this was the issue after learning more about the motherboard I have and learning it has a MemOK! feature where, on startup, the a light will either be green (if RAM is good) and red if not. It was red sure enough and replacing the RAM fixed the issue for me.
Good luck!
EDIT: Just saw Raist helped you out. Great to hear! If it happens again, check your RAM.
If it's only happening in FFXIV then it's probably not a hardware issue. You will first want to backup you HDD and then perform a clean install. Or easier buy a new HDD and do a clean install on that disk and assign you current install drive as a secondary drive. This would easily allow you to copy files over to the new HDD and if necciasry you can still boot into your original install by simply assigning you old HDD ad the primary drive and the new one as secondary.
There is no possible way for us to determine what's the problem. Perhaps you should take it to someone to look at it and fix it for you. It's extremely unlikely you are going to find the cause and solution on internet forums. You will be given a lot of of incorrect advice about power supplies or over heating or demon possession.