I generally wash my clothes in the sink and let them dry in the shower. I usually stay in cheap motels (Motel 6, Super 8, etc) that either don't have laundry facilities or where it isn't worth the headache. A 3oz nalgene squeeze bottle full of detergent is enough for a week or more. Bibs+socks+underwear get washed every night, the single bike jersey gets washed when it smells, and regular clothes get washed if obviously dirty. I do laundry as soon as I get off the bike, to allow for maximum drying time.

Bring a flexible sink stopper if you stay in cheap motels like I do; the built-in drain stopper is often completely broken or won't keep water in the sink for more than a few minutes.