After replacing the motherboard and reinstalling Windows, I still had the error. Then I found the trouble: The Das Keyboard. WTF? With it plugged in, USB Device Not Recognized every time for any other USB device. With it NOT plugged in, everything works perfectly. And this is perfectly repeatable.
So one thing to check right away, maybe first: Unplug all unneeded USB devices, and change out the keyboard and mouse.