In a large bowl, whisk together the Greek yogurt, olive oil, lemon juice, minced garlic, dried oregano, kosher salt, and black pepper to create the marinade.
Add the chicken thighs to the bowl, tossing well to ensure every piece is evenly coated. Cover and refrigerate for at least 30 minutes (or up to 12 hours for maximum flavor).
Preheat a grill or large cast-iron skillet over medium-high heat. Remove the chicken from the marinade, shaking off the excess.
Cook the chicken for 6-8 minutes per side, or until the internal temperature reaches 165°F and the outside has a nice charred crust.
Transfer the grilled chicken to a cutting board and let it rest for 5 minutes before chopping into thinner strips.
Assemble the gyros by layering the grilled chicken strips, cherry tomatoes, cucumber, red onion, and chopped parsley onto warm pita breads.
Drizzle generously with the creamy tzatziki dressing, fold the pita like a wrap, and serve immediately.