I'm originally from Greensboro, and while I wouldn't classify either of them as places not to be missed, the Bryan Park Champions Course held the USGA Public Links Championship in 2009, and The Cardinal, which is a Pete Dye course, hosts (or used to host -- I'm not sure it still exists) a relatively prestigious amateur tournament.
There's also Tot Hill Farm in Asheboro, which is one of the few Mike Strantz courses in existence.
Those are all public (or semi-private) facilities.