Add all marinade ingredients to a mixing bowl, then whisk to combine well.
Add chicken pieces to the marinade and stir to coat well. Cover and refrigerate for at least 2 hours, or up to 24 hours.
If using wooden skewers, soak them in water for at least 30 minutes, so they don’t burn too much while on the grill.
Brush grill grates and preheat grill over MED HIGH heat.
While grill is preheating, strain chicken through a sieve to remove excess marinade, and thread chicken pieces onto skewers, alternating 2 chicken pieces, then red onion, zucchini, and bell pepper, and repeat until skewers are full.
Place skewers on a wax paper lined baking sheet (for transport to the grill).
When the grill is hot, add skewers to the grill, close the lid, and cook for 8-10 minutes, flipping once or twice while cooking. Chicken should be golden brown with a little char and be cooked through.
Add chicken to a platter and if desired, sprinkle with chopped parsley and crumbled feta. Arrange some lemon wedges around the platter and serve with tzatziki sauce and warmed pita bread if desired.