Slice the chicken into evenly sized strips. Combine the olive oil, lemon juice, garlic, oregano, paprika, cumin, black pepper, and salt in a bowl. Add the chicken, toss to coat, and marinate for 10–15 minutes if time allows.
Wash and prepare the cucumber, tomato, red onion, and greens. Pat the vegetables dry and remove excess liquid from the tomato if needed.
Heat a large skillet over medium-high heat. Cook the chicken in a single layer for approximately 5–7 minutes per side, depending on thickness, until browned and the thickest pieces reach 165°F (74°C). Cook in batches if necessary.
Transfer the chicken to a clean plate and rest for 3–5 minutes.
Whisk or blend the cream cheese, feta, Greek yogurt, milk, lemon juice, parsley or dill, and black pepper until smooth. Warm gently in a small saucepan over low heat, stirring frequently. Do not boil. Add milk a tablespoon at a time if the sauce is too thick.
Warm the tortillas in a dry skillet for 15–30 seconds per side or according to the package directions, until pliable.
Place spinach or romaine on each tortilla. Divide the chicken, cucumber, tomato, and red onion among the wraps, then drizzle with the creamy cheese sauce.
Fold in the sides and roll each tortilla tightly. If desired, toast the wraps seam-side down in a dry skillet for 1–2 minutes per side.
Cut in half and serve warm with any remaining sauce on the side.