Place cucumbers in a colander; set onto a plate. Toss cucumbers with 1 teaspoon salt. Let rest to draw about liquid, about 30 minutes. Pat dry with paper towels.
Place red onion and 1 cup cold water in a bowl. Let soak until the strong onion flavor is weakened, 5 to 10 minutes. Drain and rinse.
Combine cucumbers, red onions, Greek yogurt, Kalamata olives, dill, vinegar, lemon juice, garlic, lemon zest, oregano, and black pepper in a bowl. Fold in chicken, feta cheese, and tomatoes. Cover with plastic wrap; refrigerate salad for at least 1 hour.