Using a box cheese grater, grate the trimmed cucumber on the large holes. Place in a cloth or paper towels and squeeze excess water out. Set aside.
Whisk the yogurt, garlic, lemon juice, mint, kosher salt, pepper, cumin, and cayenne pepper in medium bowl to blend. Add grated cucumbers and toss to coat. Season the mixture to taste.
Cover and refrigerate at least 1 hour. Serve with chips, pita bread or any grilled meat or burgers.