In a large stockpot cover the potatoes with enough cold water to cover plus 1 tablespoon of salt. Bring to a boil and cook for 15 minutes, or until the potatoes are fork-tender.
While the potatoes are boiling, add the chopped cucumbers and the halved grape tomatoes into a large strainer and sprinkle them with 1 tablespoon of salt. Stir. Allow them to sit in the sink and drain off their excess liquid for at least 15 minutes.
Drain the potatoes and gently stir in the olive oil, the drained cucumbers, tomatoes, red onion, and garlic.
In a large bowl, whisk together the yogurt, lemon juice, lemon zest, oregano, mint, 1 teaspoon salt, and the black pepper.
Gently stir the yogurt mixture into the potatoes until just combined.
Cover and refrigerate at least 3 hours before serving.
Garnish with additional fresh mint if desired before serving.
Notes
Store in an airtight container in the refrigerator for up to 5 days.