Sweep away any loose dried mud. Hose the area to remove more mud residue.
Mix 1 cup of laundry or dishwasher detergent in a 4-gallon bucket of water. Using a stiff scrub brush, work the detergent mix into the stained areas and scrub thoroughly. Rinse with clean water and let dry. Repeat if stains remain. You may have to repeat this several times.
Rent or borrow a power washer if the stains are especially persistent. A power washer should remove the stains quickly.
Erect fencing or another barrier to keep your dog out of muddy areas in the future.
Fill bare dirt areas of the yard with gravel, or plant grass to cover bare spots.
Wipe your dog's paws with old rags or towels on rainy days when its paws will be muddy.
Cover the area outside the dog door with a heavy rubber mat and hose off frequently to prevent setting stains.