In a small bowl, combine Greek yogurt, grated cucumber, minced garlic, lemon juice, and a pinch of salt to create the tzatziki sauce.
In a medium mixing bowl, toss the chicken with olive oil, dried oregano, salt, and pepper until evenly coated.
Heat a dry skillet or griddle over medium heat and toast each flatbread for approximately 60 seconds per side until warm and slightly charred.
Lay the flatbreads on a clean work surface and spread 2 tablespoons of tzatziki sauce over the center of each.
Divide the seasoned chicken evenly among the flatbreads, placing it on top of the sauce.
Layer the cherry tomatoes, red onion, and Kalamata olives over the chicken.
Top each flatbread with 2 tablespoons of crumbled feta cheese.
Serve immediately or wrap tightly for meal prep consumption within 48 hours.