Quick and Easy:
Always wait until the mud on your shoes is dry before attempting to clean it off. All you need to remove dried mud from shoes is a stiff scrub brush. You can also try using a nail file or other blunt instrument to tap off the dried dirt. Rub or tap off as much dirt as you can and then follow-up with a thorough cleaning.
See How to Clean Leather Shoes and/or How to Clean Sneakers for more detail.