My mother use to put whole or quartered Korean radish in a plastic container with salt and water and let it ferment. She'd skim the scum off the top every now and then, but after a while, she'd fish one out and serve it sliced, similar to the way you serve cold cucumber soup...sans fish sauce.
Now I know what you're thinking: why not ask your mother for the recipe?
She made it ages ago, and had to get the recipe from a co-worker of hers. She doesn't remember the quantities of salt, if there were other ingredients, or how long to ferment. :(
I've since seen these offered as kimchi in some Asian stores...usually floating in water flavored by the juices of the radish and a tablespoon of gochu powder that tints the juice (very slightly) red. Still, it's a four hour drive to that Asian market, and I'd much rather be able to make it at home.