In a large bowl, combine chicken pieces with olive oil, yogurt, red wine vinegar, lemon zest, oregano, and garlic. Season with salt and pepper. Let marinate for 1-2 hours, or up to overnight.
Make the tzatziki. Strain the shredded cucumber by placing it in a paper towel and squeezing the excess liquid. Combine all tzatziki ingredients in a bowl and stir to combine. Season with salt and pepper.
Make the tomato cucumber salsa by combining diced tomato, diced cucumber, red onion, olive oil, red wine vinegar, and parsley. Season with salt and pepper.
When you are ready to cook, preheat the grill to medium-high heat.
Place the chicken cubes on a skewer or in a grill pan.
Cook over medium-high heat for 3-4 minutes on each side, rotating so each side is cooked, until cooked through. Place the pita or naan on the grill for the last 1-2 minutes to warm it up and give it some grill marks.
Assemble your pitas by placing chicken cubes, tomato cucumber salsa, and tzatziki in each pita/naan. Enjoy!