In a bowl, mix Greek yogurt, lemon juice, and spices. Add chicken and coat well. Let marinate for at least 20 minutes (or up to overnight for deeper flavor).
Heat a skillet over medium-high heat. Cook the chicken for 8-10 minutes, turning occasionally, until fully cooked and lightly browned.
In a small bowl, mix yogurt, grated cucumber, lemon juice, garlic, and salt. Refrigerate until ready to use.
Lay out flatbreads and top with spinach, tandoori chicken strips, shredded carrots, shredded cheese, and a drizzle of tzatziki sauce.
Fold in the sides and roll tightly like a wrap. Serve immediately.