We don't make much use of the radish in this country. It goes into salads, sometimes it sits on the side of a plate as a garnish, and occasionally it's pickled, but that's about it. We hardly ever feature its mild, mustardy pungency by putting radish slices on buttered bread for a sandwich, the way the French do.
Our traditional radish is the smallest, mildest kind there is, harvested in spring and eaten when it's fresh and juicy. Many other radishes in Asia and eastern Europe are large and drier in texture.
In medieval Europe, huge radishes were harvested in the fall and stored in the root cellar for use throughout the winter. (The familiar red-skinned salad radish would wilt and is not worth eating in a matter of days.)
In Asia, where radishes often grow a foot long and have red, purple or even green flesh, and Eastern Europe, where a favorite kind has black skin, radishes often are cooked just like turnips, to which they are related. They're frequently pickled, and the leaves are eaten in salads. We're becoming aware of another Asian use of radishes these days: peppery, pungent radish sprouts.
Some of the most interesting uses belong to grated radish. With a little dressing, radish puree makes a nice salad, and it goes into soy-based dipping sauces for tempura and the like.
In the medieval Arab world, a little mustard and thyme would be added to grated radish to make a relish for grilled chicken (add some oil, by the way, and you have a nice salad dressing).
The most unusual use is in India, where grated radish is mixed with cornmeal to make a Punjabi corn cake called makkai ki roti. Partly it's for the flavor, but it's also because the radish makes up for corn's lack of gluten and holds the cake together as it cooks.