Combine the yogurt, olive oil, garlic, lemon juice, oregano, paprika, salt and pepper in a large bowl.
Add in the strips of chicken and toss to coat. Refrigerate for at least 30 minutes, but up to 8 hours.
Tzatziki Sauce:
In a medium bowl, combine yogurt, dill, olive oil, lemon juice, salt, pepper and sumac together.
Use a grater to grate the cucumber. Place over a small mesh sieve and let excess water drain.
Add grated cucumber into the yogurt mixture and stir until well combine. Refrigerate until ready to use.
Cooking the Chicken:
Heat a grill over medium high heat.
Remove chicken from the marinade and discard remaining marinade. Grill chicken for 4-5 minutes on each side or until it reaches an internal temperature of 165 degrees.
Assemble the Gyro:
Lightly toast or grill the pita bread.
Spread the tzatziki sauce down the center of the pita, followed with strips of chicken.
Add chopped cucumber and halved tomatoes to the pita and wrap up the end of the pita to secure. Wrap in wax paper to hold its shape if desired.
Notes
Instead of grilling the chicken, it can be cooked in the oven for 20-25 minutes at 350. Replace Greek yogurt with sour cream if you prefer. Add slices of green pepper or red onion for more vegetables in the gyro.