If you're making your own dashi, do that first. I use a simple kombu dashi, without katsuobushi for this dish (see note below concerning dashi and substitutions).

Optional - If you're using good mirin with a relatively high alcohol content, many like to very briefly boil it in order to cook off some of the sharp alcoholic flavour. Note that you will not actually boil off much of the alcohol itself by doing this. Simply heat a small saucepan over high heat, add the mirin, allow it to boil vigorously for 10-20 seconds, then remove it from the heat and allow it to cool somewhat before continuing.

Wash and halve the cucamelons and set aside. Peel the daikon and slice into very thin rounds (a mandoline makes short work of this).

Combine the dashi, rice vinegar, soy sauce, mirin, sesame oil, and ginger in a small container (preferably one with a lid). Add the cucamelons and daikon and combine thoroughly (I like to do this by putting a lid on the container and shaking everything very gently). Set aside in the fridge for 10-20 minutes.

To serve, place some shiritaki noodles in individual bowls (they don't need any cooking) and top with vegetables and the vinegar sauce. Garnish with a few radish greens if you like, and serve.