In a medium bowl, whisk 2 tablespoons of olive oil, 1 tablespoon of lemon juice, 2 minced garlic cloves, oregano, salt, and pepper.
Add chicken strips to the marinade and toss to coat evenly.
Prepare the tzatziki sauce by combining Greek yogurt, grated cucumber, the remaining 1 tablespoon of lemon juice, the remaining minced garlic clove, and fresh dill in a small bowl.
Heat the remaining 1 tablespoon of olive oil in a large skillet over medium-high heat.
Add the chicken to the skillet and cook for 8 to 10 minutes, stirring occasionally, until browned and cooked through.
Briefly warm the pita breads in a dry pan or oven until pliable.
Assemble the gyros by spreading tzatziki sauce on each pita and topping with the cooked chicken, lettuce, tomato slices, and red onion.