Prepare a charcoal grill or preheat a gas grill to medium-high heat. Meanwhile, put yogurt and mint in a small bowl and add salt and pepper to taste; stir to combine, then set aside. Toss cucumber slices in a medium bowl with remaining olive oil; add salt and pepper to taste and set aside.
Remove lamb from marinade and pat dry. Arrange on grill, fat-side down, and cook until deep golden brown, about 15 minutes. Turn lamb over and continue grilling until a meat thermometer inserted in the thickest part registers 130° for medium rare, about 15 minutes.
Transfer lamb to a large platter or cutting board and set aside to rest for 10 minutes. Cut lamb into thin slices against the grain; transfer to plates. Serve with minted yogurt, cucumbers, and olives. If desired, serve grilled skewers of grape tomatoes and warm pita bread as extra sides.
Extra idea: Turn the elements of this dish into a gyro (Greek sandwich). Finely chop the cucumber with some garlic and combine with yogurt. Spread on grilled pita bread; top with lamb and diced tomatoes.