In a small bowl, mix yogurt with dill, garlic, and 1 teaspoon lemon juice. Cover and refrigerate for at least 30 minutes to let flavors blend. In another small bowl, mix cucumber with tomatoes, green onions, and mint. Cover and refrigerate for at least 30 minutes.
Put steak in a separate shallow bowl and pour remaining lemon juice and oregano over steak. Cover and refrigerate for at least 30 minutes. Spray a nonstick saute pan with canola oil cooking spray. Remove steak from marinade, season with salt and pepper, and saute over medium-high heat for about 2 1/2 minutes on each side, or until cooked to your desired degree of doneness. Remove steak from skillet. Slice 1 side of each pita to open the pocket, being careful not to puncture the other side. Divide steak evenly among pitas and gently stuff into the pockets. Top each steak pile with 1/4 of the cucumber mixture. Drizzle 1/4 of the yogurt mixture into each pocket and serve immediately.