Sprinkle grated cucumber with 1/2 teaspoon salt in a bowl and let stand 10 to 15 minutes to draw out juice.
Place yogurt into a separate bowl. Dump cucumber and its juice onto a sturdy, dry paper or cloth towel and squeeze as much moisture as possible from the cucumber. Mix cucumber into yogurt. Add garlic, cayenne pepper, and lemon juice; mix thoroughly.
Stir dill and mint into yogurt-cucumber mixture; season with salt and black pepper. Adjust all seasonings to taste.
Cover bowl with plastic wrap and refrigerate for 3 or 4 hours (or overnight). Transfer to a serving bowl and garnish with a sprig of dill and a sprinkle of cayenne pepper for color.