I actually carry alot of my logs out of the woods to avoid this problem. Dirty logs are kind of annoying with a chain saw, but really annoying for the bandsaw mill. I also have water next to my mill so that if the logs do get muddy, i can spray them off.
When cutting very large logs, it MUST be cut completely through and you aren't going to be rolling the log to complete the cut. The forwarder operator will get pretty pi$$sed if the logs aren't cut off and it pulls fiber.