In a large bowl, combine the Greek yogurt, garlic cloves, lemon juice, olive oil, parsley, oregano, salt, lemon zest, and pepper. Whisk until smooth.
Pat the cubed chicken with a paper towel to soak up excess moisture.
Add the chicken to the marinade and stir to coat. Cover and refrigerate for 30 minutes to 4 hours. Occasionally stir the chicken to ensure it is well coated.
If you use wooden skewers, start soaking these 30 minutes in advance or overnight.
Preheat the grill to medium-high heat.
Remove the chicken from the marinade and discard the marinade.
Thread the skewers while alternating between chicken, bell peppers, onions, and zucchini in any pattern you choose.
Brush the skewers with olive oil. Grill until cooked through and the chicken reaches a temperature of 155-160ºF, rotating the skewers for even cooking.
Remove from the grill, tent loosely with foil, and allow it to rest for 10-15 minutes. The chicken will continue to rise to a final temperature of 165ºF.
Place the skewers on a serving platter and garnish with parsley.
Serve with pita bread and tzatziki.