My favorite way to eat it is to make a "fresh pickle" my mom's way,
this is only if you have fresh greens with your daikon.
Chop up fine the stems and some of the greens (the softer parts).
Chop up into small matchsticks some of the white root also (the proportions
are up to you, I like the stems & greens even w/o the root)..... salt
in a bowl, I think I use about 2 tsp to 2 cups. Rub in the salt well
with your hands, then let stand for 15 - 30 minutes. Squeeze out excess
water by handfuls. Great condiment for hot rice with some soysauce,
sometimes I add a little grated ginger also. keeps in tupperware for
about a week.
Other traditional Japanese ways are to grate it (usually served with
tempura or fish, or if you can find nameko mushrooms, or just by itself
with a little soysauce), or to stew it (it becomes much sweeter if you
stew it). You can throw in the root cut up into matchsticks when you are
making miso soup, boil a few minutes before you add the miso.
hope this helps.