Gently wash produce under cool running water, pat dry, and prepare as directed. Vegetables can be thinly sliced if shredding is difficult.
Mix together the prepared zucchini, cabbages, carrots, green onions, and radishes in a large bowl.
In a separate small bowl, mix together the mayonnaise, salt, and black pepper.
Toss the vegetables with the dressing.
Chill at least one hour before serving.