viscose towel (washcloth?)
Viscose, which is derived from cellulosic sources (wood pulp, cotton), has a highly amorphous polymer system (as well as polar polymers), making it the most absorbent fibre in common use, thus an ideal basis for a high-efficiency towel. Originally developed in 1983 by Pacific Dry Goods, but now produced by MSR, the classic Packtowl® Original (92% viscose/8% polypropylene, and pictured here) comes in a variety of sizes. It's lightweight and packs small, yet soaks up an astounding nine times its weight in water (the large size will hold a full liter of fluid), even when damp (unlike, say, terrycloth, which becomes effectively useless when wet, and — being cotton — takes a long time to dry). Further, you can release 90% of that water merely by wringing the towel out. It dries quickly as well (if still damp when you need to depart for your next destination, simply pack it in a Ziploc® bag and hang it out to dry when you arrive). Viscose towels may not look like much, or feel particularly soft (they initially resemble a piece of rather stiff felt, though become softer and more "towel-like" — and more absorbent — after each washing), but they're much more effective than other types of "travel towels". They can be machine washed (no bleaching or ironing), and air or machine dried. And they can be cut to preferred sizes, without worrying about the edges unravelling. Viscose is also biodegradable, and the brand I recommend is produced in an eco-friendly fashion.
Be aware that most products sold as "travel towels" are not viscose, but rather some sort of synthetic microfibre. Even the MSR Packtowl brand includes non-viscose forms: their Personal and UltraLite versions are made of polyester/nylon blends (85/15 and 70/30 respectively). These feel softer to the touch (which makes them more appealing, thus easily sold), but are considerably less effective when it comes to absorbing water; they are also more expensive, and slower to dry. So I strongly recommend that you avoid the hype, and get one (or more) of the viscose versions (i.e., the Packtowl Original). Remember that — for any type of towel — darker colours dry faster than lighter ones.