I'm a little late, but...
The military doesn't have any special equipment for cooling off in the desert. I was in Kuwait when it was in the 120s during the day. Most of us worked at night, when it got down in the 90s and some people put jackets on. There was no humidity. One drank loads of water and adapted. I felt seriously puffy after I'd been there a few days and realised it was because the capilaries in my skin had decided to circulate more blood out there where it could be cooled by evaporation.

Drink more water than you think you need. If it's seriously hot, you may be in the danger zone by the time you actually feel thirsty.

If you can't change the temperature, do what you can to help your body's cooling system. Wear loose, airy clothing (or none). Sit in front of a fan. Dump water over your head or water yourself while you're doing your lawn. My dad taught me how to soak a wash cloth in cold water, wring it out and then flop it on my face. The feeling doesn't last, but it sure feels good while it does!

I once knew someone who would fill a large towel with ice and suspend it from the ceiling in front of a fan. Her room always seemed a bit cooler.