Slice the cucumber half lengthwise, and scrape out the seeds with a teaspoon. Grate the cucumber using the large holes on a box grater, or the grating blade in a food processor. Toss the cucumber with 1/2 teaspoon kosher salt and place in a strainer over a bowl, or in the sink. Let rest for 10 to 30 minutes.
Use your hands to squeeze the cucumbers to press out any extra liquid, then place in a medium bowl. Add the yogurt, lemon juice, olive oil, 2 teaspoons oregano, mint and ½ teaspoon garlic. Stir well, and add pepper and any additional salt as needed (remember the cucumbers were salted at the beginning).