A bandana can do the job as well as two dozen others.
I do keep a small MSR towel/washcloth in my hygiene kit. This is the small silkier finish model and makes a better washcloth than a towel, but it is okay if wrung out-- more of a squeegee action.
I got a couple large MSR pack towels at a garage sale last year for 50 cents each, making it less painful to cut them into smaller pieces :) These are the thick felt-like ones and absorb the most water. Get a big one and split it with several others, or have a lot of spares.
Handy Wipes are pretty good, light, cheap, and available at the local grocery store. They can be cut down as well and can do most of the things a bandana can. Hmmmm, never tried one for a coarse water filter..... have to try that.
The only time I feel the need for a real towel is after swimming-- full immersion. And I'm not one for full dunking in 35F glacier-fed mountain lakes unless it is 90F out, and then I don't need a towel. For a stand-up "sponge bath" from my cut-off milk jug bucket, I really don't need much more than a small cloth to get that "squeegee" action to get most of the already thin film of water off.