What would cause an intensely itchy rash on neck, arms, & legs?

Different causes. There are different causes of this type of rash includung allergy to foods, medications, etc. Have a physician or dermatologist evaluate the rash if still present. In the meantime if you're uncomfortable you could take Benadryl (diphenhydramine) for the itch but be aware that it can cause drowsiness. Claritin is less effective but does not cause drowsiness.