In a bowl, combine olive oil, lemon juice, minced garlic, oregano, cumin, salt, and pepper to create a flavorful marinade. Add thinly sliced chicken breast to this mixture, ensuring each piece is well-coated in the marinade. Cover it with plastic wrap or transfer it into a ziplock bag, then refrigerate for at least 30 minutes or up to 3 hours for maximum flavor infusion.
While your chicken is marinating, prepare the tzatziki sauce by mixing Greek yogurt with grated cucumber, minced garlic, lemon juice, olive oil, dill (or mint), salt, and pepper in a bowl until smooth and well combined.
Once marinated, heat a large skillet over medium-high heat and add all of your chicken slices in an even layer without overcrowding them for optimal browning. Cook for about 5-7 minutes on each side or until they are fully cooked through and have developed a nice golden-brown color.
As your chicken cooks, warm your pita bread rounds either in another skillet or directly over an open flame on your stovetop until they are soft and pliable but not crispy.
To assemble your gyros, take one warmed pita bread round and layer it with cooked chicken slices first; follow this by adding chopped tomatoes, sliced red onions, chopped cucumber, crumbled feta cheese, and parsley on top of it all.
Finally, generously drizzle your homemade tzatziki sauce over everything before wrapping up your gyro tightly so nothing falls out when you take that first delicious bite! Serve immediately with extra lemon wedges on the side if desired for an added zing!