In a bowl made the chicken marinade, combine Greek yogurt, lemon juice, grated garlic, olive oil, red wine vinegar, dried oregano, salt, and pepper. Mix well.
Add the cubed chicken breasts into the bowl and toss them around to ensure they're well-coated in the marinade. Refrigerate for an hour to let the flavors meld together.
To assemble the Chicken Gyro Bowl, place rice at the bottom on each plate. Layer the cooked chicken, vegetables and feta, followed by the tzatziki sauce.