How to Make It
Stir together oil, garlic, oregano, salt, pepper, and lemon zest in a large bowl. Add chicken, and turn to coat. Cover and refrigerate 1 hour.
Preheat grill to medium-high (400°F to 450°F). Remove chicken from marinade; discard marinade. Place on unoiled grates. Grill, uncovered, until a thermometer inserted in thickest portion of meat registers 165°F, 6 to 8 minutes per side. Transfer chicken to a cutting board; drizzle with lemon juice. Rest 5 minutes or up to 20 minutes. Cut crosswise into 3/4-inch slices.
While chicken rests, place pita rounds on unoiled grates. Grill, uncovered, over medium-high until warmed and slightly charred, about 2 minutes per side. Remove from grill; spread 2 tablespoons tzatziki over each pita.
Divide lettuce, tomatoes, onion, and sliced chicken evenly among pitas.