In a bowl, mix Greek yogurt, olive oil, lemon juice, garlic, oregano, cumin, paprika, salt, and pepper. Add the sliced chicken, coat well, and refrigerate for at least 1 hour (or overnight for maximum flavor).
Grate the cucumber and squeeze out excess moisture. In a bowl, mix Greek yogurt, cucumber, garlic, lemon juice, olive oil, dill, and salt. Stir well and refrigerate until ready to use.
Heat a grill pan or skillet over medium-high heat. Cook the marinated chicken for 4-5 minutes per side until golden brown and fully cooked. Remove from heat and let it rest for a few minutes.
Warm the pita bread, then layer with lettuce, cooked chicken, red onion, tomato, cucumber, and tzatziki sauce. Sprinkle with feta cheese if desired. Wrap and serve immediately.