In a bowl, combine olive oil, oregano, garlic powder, paprika, and 2 teaspoons lemon juice. Add chicken and coat well. Let marinate for at least 20 minutes (or up to 2 hours in the fridge).
In a bowl, mix Greek yogurt, grated cucumber, garlic, dill, remaining lemon juice, salt, and pepper. Stir well and refrigerate until ready to use.
Preheat grill or grill pan over medium-high heat. Grill chicken for about 5-7 minutes per side, to 165°F(75°C). Let rest for a few minutes, then slice into strips.
Lightly warm the pita bread on the grill or in a pan. Add tomatoes, onions, lettuce, and chicken. Top with generous amount of tzatziki.
Fold the sides in and wrap with paper to hold it all together.