There is a big difference between a gopher, which is either this: http://en.wikipedia.org/wiki/Pocket_gopher , or a small ground squirrel, and a groundhog, which is also known as a woodchuck or marmot.
Woodchucks are enormous, lol. Their hole is big enough to send a dachshund down, and they can do a lot more damage. It would take a lot more to kill one of those.
Large native snake could do the trick for gophers, though--a big bull or black snake, maybe.
A Great Dane may be overkill for gophers...might want to go with a terrier, that is what they were bred for, lol.