This is perhaps a little specific, but it was vexing, so I thought I would post it in case there are other sufferers. It may be a problem that isn't specific to this particular motherboard brand/model.
I used the Startup Disk Creator in plain Ubuntu 16.04 to create a startup USB stick for Ubuntu Mate 16.04. When I attempted to boot into it using a machine with the Gigabyte GA-EP45T-UD3LR motherboard running the F5 version of the BIOS, the machine consistently hung on the BIOS screen and the USB keyboard no longer worked. A hard reset was required to get the machine to respond. Removing the USB stick while it was hung didn't restore keyboard function.
The interesting thing is that a startup USB stick of plain vanilla Ubuntu 16.04 worked fine on the problem machine, and the problem USB stick worked just fine on machines with a different motherboard and newer BIOS. That's what got me looking at issues with BIOS.
The fix was to flash the BIOS with the latest version, F10. Even though Gigabyte's site didn't report any fixes to USB issues on the BIOS versions, it did the trick. With the old version of the BIOS, the BIOS boot menu didn't recognize my USB sticks, but they were visible in the BIOS and could be selected as the primary boot device. With the new BIOS, they are detected by the BIOS boot menu (F12), and I can run the live version of Ubuntu Mate 16.04 just fine.