Whisk together yogurt, olive oil, lemon juice, minced garlic, dried oregano, salt, and pepper in a large bowl. Add chicken strips and toss to coat. Cover and refrigerate for at least 30 minutes and up to 8 hours.

Grate the cucumber on the large holes of a box grater. Place in a clean kitchen towel and squeeze out as much liquid as possible. Reposition cucumber to a dry section and squeeze again. Combine the squeezed cucumber with Greek yogurt, lemon juice, fresh dill, minced garlic, and salt. Stir, taste for seasoning, and refrigerate.

Remove chicken from the refrigerator 10 minutes before cooking. Heat a large skillet or grill pan over medium-high heat until very hot. Add a thin drizzle of olive oil. Cook chicken strips in a single layer in batches without crowding for 3 to 4 minutes per side until deeply golden and the internal temperature reaches 165°F. Transfer to a plate and tent loosely with foil.

Warm pita breads in a dry skillet for 30 seconds per side or microwave wrapped in a damp paper towel for 15 to 20 seconds until soft and pliable.

Spread a generous dollop of tzatziki across the center of each warm pita. Add a portion of chicken strips, sliced red onion, sliced tomatoes, and chopped cucumber.

Add crumbled feta and Kalamata olives if desired. Fold the pita over the filling and serve immediately.