
In a large zip-top bag or bowl, combine the chicken marinade ingredients: chicken, garlic, white wine vinegar, lemon juice, olive oil, Greek yogurt, oregano, salt, and black pepper. Mix well to coat the chicken evenly, then refrigerate for at least 2 hours, preferably 3 to 12 hours.
Make the tzatziki while the chicken marinates. Halve the cucumbers lengthwise, scrape out the seeds with a spoon, and grate them coarsely. Wrap the grated cucumber in a clean kitchen towel or paper towels and squeeze out as much liquid as possible.
Transfer the drained cucumber to a bowl and stir in the Greek yogurt and grated garlic until smooth and well combined. Chill for at least 20 minutes so the flavors can develop.
Prepare the salad by chopping the tomatoes, thinly slicing the red onion, and roughly chopping the parsley. Toss everything together in a bowl and season lightly with salt and black pepper to taste.
Preheat an outdoor grill or grill pan over medium-high heat. If using a skillet, lightly oil the pan and heat until hot.
Remove the chicken from the marinade and let any excess drip off. Cook for 2 to 3 minutes per side, or until lightly charred and cooked through. The exact time will depend on the thickness of the chicken.
Transfer the cooked chicken to a plate, cover loosely with foil, and let it rest for 5 minutes. Slice larger pieces into strips or bite-size pieces if needed.
Warm the pita or flatbread until soft and pliable. Place a portion of salad down the center of each bread, top with chicken, and spoon over plenty of tzatziki.
Fold and roll the gyros tightly. Wrap in parchment paper or foil for easier serving, then serve immediately.