Stir together the Greek yogurt, oil, lemon zest, lemon juice, garlic, oregano, 1 teaspoon salt, and ½ teaspoon pepper in a medium bowl.
Add the chicken and toss to coat. Cover and refrigerate for at least 30 minutes and up to 2 hours. (If using wooden skewers, soak them in water to cover while the chicken marinates.)
Cut the red onion into wedges and separate the layers. Stem, seed, and cut the peppers into 2- to 3-inch pieces.
Remove the chicken from the marinade, and thread onto 8 to 12 (10-inch) skewers alternately with the onion and peppers.
Preheat the grill to medium-high heat. Sprinkle the kabobs with salt and pepper. Grill, covered with the grill lid, 12 to 15 minutes, turning occasionally, or until the chicken is cooked through and the vegetables are charred and tender.
Remove the kabobs from the grill to a platter and garnish with oregano. Serve with lemon wedges, pita bread, and tzatziki, if desired.