The inspiration for these fajitas is the gyro, the Greek lamb-filled pita sandwich. Make the sauce first so it can chill while the lamb marinates.
Trim fat from lamb, and cut into thin strips. Combine lamb and next 5 ingredients (lamb through garlic) in a zip-top heavy-duty plastic bag; seal bag, and shake well to coat. Marinate in refrigerator 20 minutes.
Heat tortillas according to package directions.
Heat 1 teaspoon oil in a nonstick skillet over medium-high heat. Add lamb; sauté 6 minutes. Divide lamb evenly among tortillas; sprinkle with mint, and roll up. Serve with Cucumber-Dill Sauce.