I just got back from a week on the Voyager of the Seas. I used the valet laundry service for a pair of shorts that had dirt stains i wanted removed right away. They charge for each item. A pair of shorts $2.50, shirts $2.00, bathing suit maybe $4.00... i don't remember exactly, but it's worth it if you just have one or two items that you need laundered right away. It gets charged directly to you Sea Pass account, no taxes, no extra charges, unless you have a RUSH charge (within 24 hrs). I did mine no rush and still got it back within a day. My sister did a shirt near the end and said it took maybe a day and half.
If you don't want to use the laundry service, just make sure to hang your clothes up right after you use them, wash out swim suits in the sink and hang them up in the room (w/door open if you have a balcony) and they'll be ready to wear the next morning. Also- bring Febrese... there's a lot of smoke in the bars and clubs on board, so it helps getting the smoke smell out so you can wear them again.