In a bowl, whisk together yogurt, olive oil, lemon juice, garlic, oregano, cumin, paprika, salt, and pepper. Add chicken and toss to coat. Cover and refrigerate for at least 2 hours (or up to 8 for max flavor).
Heat a grill, grill pan, or skillet over medium-high. Brush with a little oil. Cook chicken 4–5 minutes per side, until golden and cooked through (internal temp 165°F). Rest 5 minutes, then slice into strips.
Mix all tzatziki ingredients in a bowl and chill while the chicken marinates.
Warm pitas in a dry skillet or microwave. Layer with lettuce, tomato, onion, chicken, feta, and a generous dollop of tzatziki. Fold and serve immediately.