The dish is called "namasu" in Japanese, and like eggy said, it's considered a standard side dish. (We normally have 4-5 different dishes, about 2 of them protein, along with soup and pickles with a bowl of rice for our standard meals -- extremely well-balanced way to eat, with heavy emphasis on vegetables.)

Daikon and Carrot Namasu

500 grams daikon (Japanese turnip -- I call them turnip to differentiate them from "kabu" which is shaped like radishes)
50 grams carrot
3 Tablespoons Japanese rice vinegar
1 Tablespoon sugar
1 Tablespoon dashi (smoked bonito stock)
1/2 teaspoon salt
Toasted white sesame seeds to taste

Julienne daikon and carrot into match-sized pieces, and sprinkle some salt to soften. Combine rice vinegar, sugar, dashi, and salt in a separate container. Squeeze the moisture from daikon and carrots, and add them to the vinegar mixture. Sprinkle some toasted sesame seeds and serve, chilled.

If you need dashi recipes, I posted them in response to Julie C. on the vegetarian site not too long ago -- or you can use the dashi powder/pellets (I don't use them unless I'm desperate because they contain MSG).