Core the apples and peel the beets. Grate the apples, beets and carrots with a box grater or with the grating attachment of a food processor. Put grated vegetables in a large bowl.
Put remaining ingredients in a jar and shake well. This is more dressing than you will need. Pour in just enough to coat the salad, but not enough to drench it. Mix well. Save the rest of the dressing in your fridge for other salads.