Whisk olive oil, lemon juice, garlic, oregano, cumin, paprika, salt, and pepper in a bowl.
Add chicken and toss to coat well. Marinate 15–30 minutes at room temperature.
Preheat a grill or grill pan to medium-high. Oil the grates lightly.
Grill chicken 5–6 minutes per side until cooked through and 165°F inside.
Rest chicken 5 minutes, then slice thinly across the grain.
Stir yogurt, cucumber, dill, vinegar, olive oil, and a pinch of salt in a bowl.
Warm pitas on the grill 30–60 seconds per side until pliable.
Fill pitas with lettuce, sliced chicken, tomatoes, onion, and feta.
Spoon yogurt sauce over the top. Add hot sauce if desired and serve.