Linen.
Cotton is good and costs less, but when I have had linen it was far cooler than anything else I’ve ever worn.
Velour, if you believe in Zapp Brannigan’s guide to making love at a woman.
Synthetics can be awesome. You don’t need to buy sports clothing. Most travel clothing suppliers make warm-weather friendly synthetic clothing that looks relatively normal. It can get pricey, but REI Outlet and Sierra Trading Post usually have some cheaper options. Keep the fit loose and breezy, coverage high (loosely covered is cooler than bare), and colors light. Failing that, just remember to never, ever, wear jeans.
I’ve had silk long underwear, and it was the least warm long underwear I’ve ever worn. It was comfortable.
In this very forum I read that aviators wore silk scarves to filter exhaust from their plane engines. The scarves were long so that when one part got dirty the pilot could shift the scarf to put a clean part over his mouth many times during a flight.
Past tense unnecessary.
It depends on what sort of clothes; if we’re talking suits, then something like seersucker is probably your best bet.
If we’re talking casual, I’d go with one of the new treated polyester golf shirts and a pair of lightweight khakis.
Cloth of gold is cool…but pricey!
Nothing cooler IMHO than a pair of summer weight wool grey slacks and a Tom Ford white shirt. + Mosley Tribes glasses. 
Loose fitting cotton (linen is itchy).
OmniFreeze Zero. I imagine it’s pretty pricey, though.
linen/cotton, or if you’re doing stuff merino wool
or maybe a jacket filled with steel ball bearings?
The best thing when discussing this every summer is I get to say “wick” a lot.