Slice the peeled cucumbers in half lengthwise. Using a teaspoon, scrape out the seeds and discard. Sprinkle the cucumber halves with the salt and set aside for fifteen minutes. Rinse off salt and dice into small pieces.
Combine the diced cucumbers, yogurt, dill, garlic, and olive oil and stir until mixed. Place mixture into the bowl of a food processor and puree; leave it chunky. Season with salt and pepper to taste, and chill until very cold, at least 1-2 hours. When ready to serve, garnish with sliced radishes, chopped cucumbers, and a sprig of dill.
This soup can be prepared ahead and stored in the refrigerator for 1-2 days.