In a medium bowl, whisk together the olive oil, lemon juice, minced garlic, oregano, salt, and pepper to create the marinade.

Add the cubed chicken to the marinade, toss to coat thoroughly, and cover. Refrigerate for at least 20 minutes, or up to 4 hours for best flavor.

Heat a large skillet or grill pan over medium-high heat. Add the marinated chicken pieces, ensuring not to overcrowd the pan (work in batches if necessary). Cook for 6-8 minutes, turning occasionally, until the chicken is golden brown and cooked through (internal temperature reaches 165°F/74°C).

While the chicken cooks, warm the pita breads according to package directions (usually wrapped in foil in the oven or quickly heated on a dry skillet for 30 seconds per side).

Assemble the pitas: Lay open a warm pita. Spread a generous layer of tzatziki sauce inside. Top with the cooked chicken, followed by the halved cherry tomatoes, diced cucumber, sliced red onion, and a sprinkle of crumbled feta cheese.

Fold the pita around the filling and serve immediately. Offer extra tzatziki on the side for dipping.