On a tour? Make sure you carry extra water as archaeological sites get very hot and the refreshment stands can run out of aqua.
Slather on that sunscreen, wear hats, and eat a few olives now and then for salt.
Buy or take a plastic spray bottle filled with water to mist yourself. Local alternative? A Greek sea sponge kept just damp with water.
Sunstroke is a real danger - keep in the shade when possible. If you stop sweating, that's a major danger sign.
See if you can visit sites at opening, when it's cooler. Late afternoon is too late - most of them close at 3pm. In summer, however, many important sites are staying open into the evening. Check locally; the hours may be longer than written sources indicate.
If your lodgings aren't air conditioned, get out into one of the parks, down to the water, or up on a hill. At the very least, find a shady taverna. The air will be fresher, you'll feel better, and the heat won't be as oppressive when you can see all of Greece sweating along with you. Go to one of the open-air cinemas - it doesn't matter what they're showing.
Skip heavy, hot foods. Indulge in cold drinks and cold desserts like ice cream - for once in your life, they may actually be good for you! Live off the mezedes (appetizers, often room temperature or cold). Tzatziki sauce is your best friend, made of cooling cucumber and yogurt.
High temperatures in Greece can bring forest fires. Remember, if air quality is already poor from fire-related pollution, not always obvious to the naked eye, the high temperatures will hit you harder. Ask your doctor if you should consider taking extra anti-oxidant vitamins or "anti-smog" compounds to help you clear your system.
