Combine olive oil, lemon juice, garlic, oregano, thyme, and cumin in a large bowl. Season with salt and pepper. Add chicken pieces and toss well to coat chicken. Cover and transfer to refrigerator for at least 1 hour, up to 4 hours.
Heat grill to medium-high. Thread chicken, onion, and peppers onto skewers, distributing ingredients evenly.
Brush grill lightly with oil. Grill kebabs for about 10 minutes, turning once, until chicken and veggies are cooked through and nicely charred. Transfer to serving platter and top with olives and parsley. Serve with tzatziki sauce.
Quick Tzatziki Sauce:
Combine 1 cup Greek whole milk yogurt, 1 English cucumber (seeded, finely grated, and drained), 2 cloves minced garlic, 1 teaspoon lemon zest, 1 tablespoon lemon juice, and 2 tablespoons chopped fresh dill in a medium bowl. Season with salt and pepper. Cover and chill.