Slight update to my earlier post. I opened up the case again this morning and verified that no cables had accidentally gotten jostled loose when I installed the new GPU yesterday. Everything seemed snug as a proverbial bug, so I don't think that's the issue.
I have two theories at this point; not sure how valid they are:
1) Something's gone off with the PSU. I noticed a slight popping sound when I plugged the power cord back in this morning (even though yes, the PSU was switched off), if that makes any difference. Only way to test this theory really is to haul it into the shop and have them test it. (Though, they tested it for me when the LEDs on the front of the case went out back in January/February and said it was fine. And yes, I trust said shop; I know the owner.)
2) I screwed the pooch when I cleaned out the old AMD/ATI drivers with Driver Sweeper. Maybe it did something it shouldn't have. I could roll back via System Restore, but unfortunately the latest restore point I have is prior to installing the new GPU. (For some reason Windows created a restore point when I installed Silent Hill 4, but not when I uninstalled/reinstalled video card drivers. WTF, Windows.) I'm not sure rolling back that far is a smart idea given the GPU change.
Any thoughts? Once again, this has only happened once so far, but even once is one time too many for me.