Whats the best way to clean a large area rug?
Many commercial rug shampooers will do the trick. Since you say there are no urine stains or smells, the dog smell should be gone by the time the rug dries, after you shampoo it. Try a rental machine yourself first. I've rented them before, but can't remember specific brands. Shampoo it thoroughly with the machine, then let it dry, preferably out of doors. Try laying it out on a picnic table, or even on the grass to dry. Drying it outdoors will help dry faster and help remove any lingering odors. If this doesn't work, call a professional rug cleaner, or throw the thing away.
Karen's answer would be best, just put it in the yard and soak the crap out of it.. but if you don't have a yard, the bathtub will do. Put it in, put some detergent in, (and maybe the salt to help keep the dye from going anywhere) swish it around a lot and maybe let it soak, then drain and refill the tub until you feel it's rinsed.