What actually works REALLY well as a cooling cloth is a flat cloth diaper. (Not the kind with the heavy panel down the center, just the flat piece of cotton birdseye.) They are bigger than a facecloth and much lighter than a towel, also thin enough to tie around your neck, though they are super-absorbent.
No one who sees it will know it's a diaper; it justs looks like a large white handkerchief. They make good dustcloths, too; I buy a pack every so often and keep 2-3 solely for cooling purposes.
Drink a lot of water, and use a good sunBLOCK, lots of it. Reapplying every hour will actually help your skin literally stay cooler (part of the hot sensation is a side-effect of your skin getting cooked by UV). Misting fans are great. Also, carry some ibuprofen, etc. in case you get a headache; you'll want to stop it before it gets serious. If anyone comments that you are looking oddly pale, get indoors and wet yourself down immediately; that is a warning for heatstroke.