Add the potatoes to a pot and cover them with cold water. Bring them to a boil over high heat, reducing to a simmer. Continue cooking the potatoes for about 20-25 more minutes, checking their doneness until they're fork tender (easily pierced, but not yet completely mushy, with no raw taste).
Drain the potatoes and leave them in the strainer to cool completely. If you're in a pinch you can run them under cold water to speed up the process. Or make them ahead of time, cool, and store int he refrigerator until ready to finish the salad.
Add the remaining ingredients, except for the cucumber, to a large mixing bowl-stirring to evenly combine the sauce.
Shred the cucumber in the bowl of a food processor with the grating attachment. Spread the shredded cucumber evenly out between several layers of paper towels and press down firmly to release excess moisture. Repeat 2-3 times.
Stir the shredded cucumber into the sauce until evenly incorporated. Add the potatoes to the mixing bowl, gently tossing to coat.
Salt and pepper again, if desired. Serve immediately, or cover and refrigerate until chilled through. Sprinkle with crumbled feta and mint leaves if you want a gourmet garnish.