This site has a wide selection of evaporative and solid-liquid phase change cooling apparel. I don't have any experience with this site - the place where I bought mine is apparently out of business - I just put the link up because they had so many different choices you could see.

It occurred to me that in the desert, an evaporative vest would probably work just fine at bicycling speed. In the humid east, more air flow is needed.

I haven't had good luck with the bandana things, but I know some people really like them. To me, they just insulate my neck, and the coolness from the evaporation on the outside layers never reaches the inner surface. Maybe, again, ambient humidity and reduced air flow from my short neck are the reason for that.