In a mixing bowl, whisk together 3 tablespoons of olive oil, 2 tablespoons of lemon juice, 2 minced garlic cloves, oregano, salt, and pepper. Add the chicken breasts and marinate for at least 15 minutes.

In a separate bowl, combine 1 cup of Greek yogurt, the remaining 2 tablespoons of olive oil, 1 tablespoon of lemon juice, 2 minced garlic cloves, grated cucumber, dill, salt, and pepper. Squeeze the cucumber to remove moisture, mix well, and set aside.

Heat a grill pan over medium-high heat. Cook the marinated chicken strips for 2-3 minutes on each side until golden brown and cooked through, reaching an internal temperature of 165°F (74°C). Let them rest.

Warm the whole wheat pita pockets in a dry skillet over medium heat for 1-2 minutes per side until soft.

Spread tzatziki sauce on the warm pita, add chicken strips, cherry tomatoes, red onion, and olives. Wrap the pita to enclose the filling.