Whisk together lemon zest, 1 tablespoon plus 1 1/2 teaspoons lemon juice, half the garlic, the oregano, 1 tablespoon plus 1 teaspoon oil, and 1/2 teaspoon salt. Add chicken, and toss to coat. Refrigerate, covered, for 45 minutes.
Meanwhile, stir together remaining lemon juice and garlic, the cucumber, yogurt, and 1/4 teaspoon salt. Refrigerate, covered, until ready to serve.
Heat grill to medium-high. Thread chicken, tomatoes, and onion onto 4 metal skewers. Brush with remaining 2 teaspoons oil, and season with pepper.
Grill skewers, turning, until browned on all sides and cooked through, about 8 minutes. Grill pitas until charred, about 2 minutes per side.
Serve skewers with charred pitas and cucumber-yogurt sauce. Garnish with oregano leaves.
Make ahead: Cucumber-yogurt sauce can be refrigerated overnight.