Well, here is the summary in case it can help someone else in the future:
The Gigabyte 970A-D3, "Ultra Durable", Rev 3.0. works correctly with 32-bit Linux Kernels but USB and Networking don't work (with the default bios settings) with 64-bit Linux Kernels. This appears to be true up to at least Kernel 3.7, possibly later.
Some other people have experienced similar issues with other motherboards with similar chipsets.
In my case, changing the bios setting of IOMMU from the default of disabled to enabled seemed to fix the problem.
I'm going to report the problem to launchpad, hopefully they will pass it upstream and get it changed so that 64-bit kernels will work with this board (with its bios defaults), or give useful error messages that will direct people to the IOMMU settings.