I always dry my dishes right away and put them up, never let them air dry.  I use a synthetic car drying towel (Absorber, Shamwow, pick a name...) cut in half.  Cheap and you don't need (or want) them to by 'dry'.  I keep it in a quart ziploc bag in my wash kit to keep it clean and from drying out.  Much cheaper than a 'camp towel'. 

I use the other half to dry me or sometimes dry the dew/rain off the tarp or tents so they pack a bit dryer than just shaking them off.

pd