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).
And if you are as fond tzatziki as much as I am, you may want to make extra to serve up with pita chips the next day, or perhaps to dollop on a piece of grilled chicken or a chop. Leftover roasted meat thinly sliced and piled into a pita with tzatziki makes a great makeshift gyro (lamb, beef, pork, or yes, turkey!). Tzatziki is best eaten within a day of making, as the mixture can start to become a bit watery from the cucumbers.