I'm going on year #5 of radishes and turnips with no problems. I have just one 1.5 acre plot. It never gets tilled or sprayed, just continually over-seeded with the next seasons crops. It contains perennials like ladino, medium red, and yellow sweet clover, and alfalfa, and then gets buckwheat, sunflowers, oats, winter rye, radishes, and turnips broadcasted at their appropriate times. It also has an abundance of purple coneflower, bachelor's buttons, Shasta Daisy, bee balm, anise hyssop, milkweed, and hollyhocks, in certain areas.
With lots of diversity and something always growing, disease and insects have become a non factor. Generally, disease and insects are a symptom of having unhealthy plants, which is a symptom of having unhealthy soil.