Slice the cucumber in 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 ½ 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, oregano, mint, and garlic. Stir well, and add pepper and any additional salt as needed (don't forget that the cucumbers were salted at the beginning!).
Notes
Tzatziki is best eaten within a day of making, as the mixture can start to become a bit watery from the cucumbers.