The traditional sauce for gyros is tzatziki (Greek name), or cacik (pronounced jajik) in Turkey. I've lived in both countries and the recipe is the same. This is a traditional recipe.
First, pick up a large container of plain yogurt, full fat, low fat, or fat free. They all work. Line a collander with plain white paper towels, dump the yogurt into the middle, top with a paper towel. Set the collander over a bowl to catch the whey and set it in the refrigerator to drain. The longer it drains, the thicker it will be. If you let it go long enough, you'll have yogurt-tart cream cheese!
Peel a cucumber, cut it in half, stem to blossom end. Scoop out the seeds with a teaspoon and discard. Grate cucumber, place it in a paper towel and squeeze out as much moisture as possible. Place drained yogurt in a bowl, add the cucumber and two or three cloves of chopped/mashed garlic. I simply use a garlic press. Add a scant teaspoon of salt and 2 tablespoons of good olive oil. Stir to blend and refrigerate for a couple of hours while the flavors merge. That's it! No mayonnaise, and for pity's sake, do NOT use sour cream! A local Greek restaurant does, and it's a travesty! Tzatziki will keep in the fridge for four days or so, but mine is always gone a lot quicker than that!
Oh, and fresh garlic has a nice bite. Garlic powder does not. I prefer the fresh. It's not unheard of to add a bit of dill weed or even spearmint. Or both. Fresh dill is better than dried. But the basic recipe is pretty close to universal in both Greece and Turkey. It's delicious!
(This is from a message board in which someone asked what the white sauce on a gyro was).