As a line medic for a light infantry unit in Iraq and an aid station medic in Afghanistan, I have some experience with trying to stay cool as well as seeing what happens to those who don't stay cool/hydrated enough. Being on our feet isn't exactly the same as being on a bike, but I can still tell you what worked for me.
First of all, stay hydrated! Your body's cooling system works by losing water. You need to constantly drink liquids to keep your body running. Caffeine and alcohol are both diuretics, which mean they cause you to piss out more than you take in. Avoid these... At least before a particularly long hot leg of your trip. And remember, proper hydration starts the night before, not the morning that you leave.
I've seen many, many cases of heat exhaustion... Not a fun thing to go through, especially if you find yourself on your bikes a long way from anything. Heat exhaustion can rapidly progress to heat stroke. I've seen only one case of heat stroke. Very scary. The man was very lucky to live through it. I know I am being a bit dramatic, but I just want to emphasize how important staying hydrated is... so along with that, make sure to carry plenty of extra water for just in case. Even if you don't have room for it, make room.
Second of all, any of those gel-filled cooling vests, neck bands, etc. are spectacular. The cooling they provide is really pretty modest, but every little bit helps, and it really makes a big difference by the end of the day. They work just like a wet shirt, but they last a lot longer. They would last for several hours on a mission while walking. I don't know how long they would last on a motorcycle with the wind beating against them though.