I bought one of those Headsweats 'do rags' a couple of years ago, meant to be worn under the cycling helmet. Made out of a moisture wicking fabric to aid in cooling.
If I was to riding the AZ sun, I'd consider a DaBrim though...
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'75 Fuji S-10S bought new, 52k+ miles and still going!
'84 Univega Gran Tourismo
'84 Univega Viva Sport
'86 Miyata 710
'90 Schwinn Woodlands
Unknown brand MTB of questionable lineage aka 'Mutt Trail Bike'
Plus or minus a few others from time-to-time