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Using a food processor, finely shred the cucumber.
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Wrap the shredded cucumber in cheesecloth or a clean dishtowel and wring unit all of the water is out of the cucumber.
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Place cucumber in a fine meshed strainer (or over another piece of cheesecloth) over a bowl and allow to rest.
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Using a food processor, finely shred the radishes.
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Wrap the shredded radishes in cheesecloth or a clean dishtowel and wring until the water is out of the radishes.
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Add the radish shreds to the strainer (or piece of cheesecloth) that holds the cucumber, allowing them to continue draining while you finish the rest of the dip.
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In a large bowl, combine remaining ingredients and mix well.
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Add cucumbers and radishes to dip, mix well.
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Place into refrigerator for 1 hour to allow flavors to combine.
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Stir well, garnish with radish and cucumber shreds or thinly sliced scallions and serve.