Quarter fennel bulbs lengthwise and cut out and discard most of core, leaving just enough to keep the quarter intact. Cut fennel lengthwise into paper thin slices. Transfer to a large bowl.
Cut radishes crosswise into very thin slices (slightly thicker then fennel slices). Add to bowl.
Cut, peel, including all white pitch, from oranges with a small sharp knife. Working over a small bowl, cut segments free from membranes and squeeze juice from membranes into bowl. Transfer segments to large bowl with fennel and radishes.
Using the salt, mash the garlic clove into a paste. Add that to bowl with orange juice. Add black pepper, olive oil and cider vinegar. Whisk until well blended. Pour over fennel, radishes and orange sections. Toss to coat all pieces.