Getting clothes cleaned while traveling can often be a challenge, a pain, or one of the fun things you do on the road. We've visited laundromats in foreign countries and had lots of fun figuring things out often with the help of the locals who I am sure we entertained. If you are planning on more than a week of travel, chances are you will want to schedule some time to wash a few items. It makes for a lighter suitcase if you don't pack clean clothes for every single day.
If you aren't up for the challenge of a local foreign laundromat, you may opt for hand washing in the sink in which case you need to plan for a couple of days in one spot so that your clothes will dry. Pack lighter weight fabrics and plan to layer if the weather is cool. The lighter fabrics will dry faster for you but still you will need a good twenty-four hours to be sure they get dry hanging in the bathroom or a closet. Be sure to pack one of those flat rubber stoppers to place over the drain. If you don't want to pack a little detergent, you can always use some shampoo if you are just trying to refresh your things and not get out mud or grease.
Or if you are really into washing your clothes you could try a new gadget I happened across called the Scrubba. It's advertised as the smallest traveling washing machine. It's a reusable pouch that has a rough texture inside (think wash board). You fill it with water and detergent, work the air out, seal it and knead it back and forth until the clothes get clean. It costs between $45-50 USD at Amazon. While it has some good reviews, I can do a lot of laundry by hand or even at the local laundromat for that much money. Plus I'd have to pack it even if it is small.