First thing is to leave anything made of cotton at home. Coolmax or other synthetic blends dry much faster. Plus, when worn a little damp, the clothes don't seem as clammy and dry anyway once you start riding. Also, black garments heat up faster in the sun and thus dry more quickly than light colored ones. But lower temps and high humidity are not never helpful when doing laundry on the road.

If the weather is cooperative, I do my washing in a five gallon zip lock bag with biodegradable soap. All rinsings go into toilets or are poured well away from surface waters. I then hang everything with clothespins on a plastic clothes line. If I'm lucky, everything dries before dark, if not - usually not - I hang clothes from parachute cord strung up inside my tent. If still damp the next morning - usually is - I either wear the clothes anyway or spread them out inside the mesh external pockets of my panniers and flag them off the rear rack of my bike.