The trick I learned is to remove all the pouches off the vest and then good over with mild detergent and water. I prefer to wash it by hand - takes more time, but i know it will get really clean. For heavier grime, a good spray with the hose should remove most of it.
Something to consider when putting pouches on a MOLLE vest - use "speed clips" instead of weaving the webbing. Speed clips make so much easier to add and remove pouches. Blackhawk makes a great speed clip - check it out below: