Assuming you're talking about a rug that can be taken outside and beaten:
First, vacuum the rug to remove any loose dirt and debris. If the rug is really dirty, you may want to shampoo it using a carpet cleaner.
To remove any remaining dirt and grime, mix a solution of one part vinegar to three parts water. Using a sponge or brush, scrub the solution into the rug and then rinse with clean water. You can also use this solution to spot-clean any stains.
Let the rug air dry completely before putting it back in your mudroom.