Slice the cucumber in half. Use a spoon to scrape out the seeds, then grate it onto paper towels. Squeeze the cucumber in the paper towels to remove the excess moisture, then set aside.
In a bowl, stir together the yogurt, cumin, coriander, cayenne, lemon juice, salt and pepper. Add the cucumber and stir to combine, then stir in the mint. Taste and adjust the seasoning as you desire.
Cover and refrigerate for a couple of hours to allow the flavors to combine. Serve as a dip with pita wedges.