1. Stir together 1/2 cup yogurt, 2 tablespoons juice, 1 tablespoon garlic, and oregano in a resealable plastic bag. Add chicken, reseal bag, toss to coat evenly with marinade and chill, loosely covered, for at least 15 minutes and up to 1 day.
2. Whisk together remaining 1 cup yogurt, 2 tablespoons juice, 1-1/2 tablespoons garlic, and 1/4 cup cucumber for dressing; chill, covered, until ready to serve.
3. Preheat outdoor grill or a grill pan and lightly oil rack or pan.
4. Brush cut sides of pita evenly with oil and grill on rack over moderately high heat, turning once, until golden brown, and slightly crisp, about 2 minutes. Transfer to a rack to cool. Remove chicken from marinade, discarding excess marinade, and grill with bell pepper over moderately high heat, in batches if necessary, turning until golden brown and cooked through, about 3 minutes for pepper and 6 minutes for chicken. Transfer pepper and chicken, as grilled, to a plate.
5. Arrange 1 pita half on each of 6 plates and top evenly with tomato, pepper, feta, chicken, olives, and remaining cucumber. Drizzle with some of dressing; salt and pepper to taste. Top with remaining pita half. Serve with extra dressing and romaine leaves on the side. Makes 6 servings.