In a bowl, toss the chicken strips with 2 tablespoons olive oil, juice of one lemon, 2 minced garlic cloves, 1 teaspoon oregano, salt, and pepper.
Heat a large skillet over medium-high heat. Add the seasoned chicken and cook for 6 to 8 minutes, stirring occasionally, until lightly browned and cooked through.
While the chicken cooks, grate half the cucumber and slice the remaining half. In a small bowl, stir together the Greek yogurt, grated cucumber, remaining minced garlic clove, dill, and juice of half a lemon to make the tzatziki.
In another bowl, combine the sliced cucumber, diced tomatoes, red onion, feta, remaining tablespoon of olive oil, remaining lemon juice, and remaining teaspoon of oregano. Toss gently to make the cucumber salad.
Warm the pita breads in a dry skillet or microwave for 20 to 30 seconds each. Fill each pita with cooked chicken and a spoonful of tzatziki. Serve the cucumber salad on the side.