Trim the ends of 1 medium English cucumber, peel most or all of the peel, and cut it in half lengthwise. Scoop out and discard the seeds. Shred cucumber using the large holes of a box grater, then lightly chop the shredded cucumber. Place the shredded cucumber on a clean dishtowel, and squeeze out the excess liquid.
Whisk 1 cup plain Greek yogurt, 2 Tablespoons olive oil, 2 Tablespoons minced fresh dill, 1 small clove garlic, 1/2 teaspoon kosher salt, and 1/4 teaspoon freshly ground black pepper together in a small bowl or large measuring cup. Stir in shredded cucumber. Cover and refrigerate for at least one hour. Refrigerate leftovers in an airtight container for up to 3 days.