Place the diced cucumber in a colander and sprinkle with the salt, tossing to mix well. Set the colander in a bowl, cover with plastic wrap and allow to drain for 3 to 6 hours. Line a sieve with cheesecloth and place the plain yogurt on top of the cheesecloth. Place the sieve over a bowl and allow to drain for 3 to 6 hours.
Pat the drained cucumber dry and transfer to a large bowl. Add the yogurt and stir to combine. Add the olive oil, 1 tablespoon lemon juice, garlic and mint, mixing gently until thoroughly blended. Season to taste with additional lemon juice, salt and freshly ground pepper.