In a bowl, combine olive oil, lemon juice, minced garlic, oregano, cumin, salt, and pepper. Whisk to create a marinade.
Add the chicken cubes into the marinade and stir well to coat. Cover and let it marinate in the refrigerator for at least 30 minutes (up to 2 hours for more flavor).
Preheat your grill or grill pan over medium-high heat. Thread the marinated chicken onto skewers, leaving some space between pieces.
Grill the chicken skewers for 10-12 minutes, turning occasionally, until fully cooked and charred in spots (internal temperature should reach 165°F).
While the chicken is grilling, warm the pita breads on the grill for 1-2 minutes per side until slightly toasted.
To assemble, carefully remove the chicken from skewers and place it inside each warm pita.
Top the chicken with cherry tomatoes, cucumber, red onion, a dollop of Greek yogurt, and sprinkle with feta cheese.