In a medium bowl, combine olive oil, lemon juice, minced garlic, dried oregano, ground cumin, smoked paprika, salt, and black pepper to create the marinade.
Add the thinly sliced chicken breast to the marinade and toss to coat evenly. Cover and refrigerate for at least 15 minutes.
While the chicken marinates, prepare the tzatziki sauce by mixing Greek yogurt, chopped fresh dill, lemon juice, diced cucumber, and salt in a small bowl. Stir well and refrigerate until ready to use.
Preheat a grill pan or skillet over medium-high heat.
Remove the chicken from the marinade and cook on the grill pan for 5-7 minutes per side or until fully cooked and internal temperature reaches 165°F (74°C).
Warm the whole wheat pita breads in a dry skillet or oven.
To assemble each gyro, spread 2 tablespoons of the tzatziki sauce onto the center of a pita bread.
Add a generous portion of grilled chicken slices on top of the sauce.
Top with shredded romaine lettuce, tomato slices, and red onion slices.
Fold the pita over the fillings and serve immediately.