Mix the dressing ingredients in a small mixing bowl and refrigerate until ready to use.
Bring a large pot of salted water to a boil over medium-high heat. Carefully add the potatoes and eggs and let boil for 8-10 minutes.
Remove the eggs from the water and continue the potatoes boiling for additional 5-10 minutes, or until fork-tender.
While the potatoes are boiling, run the eggs under cold running water to stop cooking, remove the shell and set aside.
Meanwhile, prepare the remaining ingredients. Slice the cucumber halfway lengthwise. Using a spoon, scoop out the seeds avoiding the flesh and chop. Slice the radish and finely chop the green onions.
Chop the eggs into bite-size pieces. Once the potatoes are done, remove from the heat and allow them to cool for about 10-15 minutes.
Chop them into 1/4-inch thick rounds and place them into a large bowl. Add cucumber, radish, green onions, eggs and a generous pinch of salt to the potatoes.
Pour the dressing all over the veggies. Mix well to evenly coat. Cover the bowl with cling film, and refrigerate to enhance the flavors for at least 15 minutes.
Sprinkle some more black pepper and enjoy.