Peel the cucumber (optional) and cut in half lengthwise. Scoop out seeds with a spoon.
Grate the cucumber using the large holes of a box grater.
Place grated cucumber in a clean kitchen towel and squeeze firmly to remove as much liquid as possible.
In a medium bowl, combine the yogurt, drained cucumber, minced garlic, lemon juice, chopped dill, olive oil, and salt.
Stir well until completely combined.
Cover and refrigerate for at least 1 hour to allow flavors to develop.
Before serving, drizzle with a little more olive oil and garnish with fresh dill sprigs.