Thanks for the info, Netkas.
I was wrong and my other Mac actually has OS 10.9.5 on it. Still, I tried copying its AMD and ATI extensions to my Mac Pro, but that didn't help anything; I just got a black screen. So I decided to copy back the 10.10.3 driver files to my extentions folder (I had saved them); then I got back an image on my display, but I lost hardware acceleration. No idea why. There must have been a simpler way to fix this, but as I said I don't know much about playing around with extensions, so I reinstalled Yosemite from scratch and then copied my user files from a backup. Fun times!
I don't know if the 10.10.2 drivers would have worked any better. The Yosemite installer from the Mac App Store is already at 10.10.3 so I don't know how to get a hold of a 10.10.2 system to copy the files.
In any case, it would seem to me that even if the 10.10.2 files did work on 10.10.3, this would not be a good long term solution anyway. If the 10.9.5 drivers don't work on 10.10.3, I suppose there's a good possibility the 10.10.2 drivers won't work once Apple releases 10.11, and then we're back to square one.
So once again, thanks Apple.