In a small bowl, whisk together olive oil, fresh lemon juice, minced garlic, dried oregano, ground cumin, kosher salt, and black pepper.
Place chicken breasts in a resealable plastic bag or shallow dish and pour the marinade over, ensuring each piece is well coated.
Refrigerate for at least 30 minutes, preferably overnight.
Preheat a grill or grill pan over medium-high heat. If using a grill, lightly oil the grates.
Remove the chicken from the marinade, discarding the excess.
Cook the chicken for 6-7 minutes on each side or until fully cooked. The internal temperature should reach 165°F (74°C).
Once cooked, let the chicken rest for a few minutes before slicing on a cutting board.
Lay out a tortilla or pita bread and place sliced chicken in the center.
Add slices of fresh cucumber, red onion, halved grape tomatoes, and crumbled feta cheese.
Drizzle 1 tbsp of tzatziki sauce over the fillings.
Fold the sides over the fillings to create a wrap.
Slice the wrap in half for easy serving and enjoy.