Whisk the olive oil, lemon juice, garlic, oregano, salt, pepper, and paprika in a large bowl.
Add the chicken and toss to coat. Cover and refrigerate for at least 30 minutes or up to 12 hours.
Thread the chicken onto skewers, leaving a little space between the pieces.
Make the Tzatziki:
Stir the yogurt, grated cucumber, lemon juice, garlic, and parsley together. Season lightly with salt and refrigerate until serving.
Grill and Serve:
Heat a grill or grill pan over medium-high heat. Grill the skewers for 10 to 12 minutes, turning often, until browned and the chicken reaches 165 F (74 C).
Warm the pita. Fill each with chicken, tzatziki, tomato, cucumber, red onion, and lemon. Serve immediately.
Notes
Chicken thighs stay juicy and are ideal for skewers.
Squeeze the cucumber dry so the tzatziki stays thick.
The chicken can also be cooked without skewers in a hot skillet.