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Combine lemon zest, lemon juice, 3 tsp. oil, oregano, 2 tsp. garlic, and crushed red pepper in a large bowl. Add chicken and toss to coat. Marinate in the refrigerator for at least 1 hour or up to 4 hours.
Meanwhile, toss grated cucumber with ¼ tsp. salt in a fine-mesh sieve. Let drain for 15 minutes, then squeeze to release more liquid. Transfer to a medium bowl. Stir in yogurt, mint, dill, ground pepper, and the remaining 2 tsp. oil, ¾ tsp. garlic, and ¼ tsp. salt. Refrigerate until ready to serve.
Preheat grill to medium-high.
Oil the grill rack (see Tip). Grill the chicken until an instant-read thermometer inserted in the center registers 165 degrees F, 3 to 4 minutes per side.
To serve, spread some of the sauce inside each pita half. Tuck in the chicken, lettuce, red onion, tomatoes, and sliced cucumber.