Hi Sarah,
Actually tics are not uncommon in children of that age. Especially boys it seems. It has something to do with their brain development. In fact, there were a number of children in my children's early grades who had transient tics and without diagnoses or treatments even, their tics were never to be seen again after several months. Unlike our own.

DSM-IV - Diagnostic criteria for Transient Tic Disorder

My primary concern would be the anxiety and sleep issues that you mentioned despite you saying his tics have gone after elimination diets etc..

Tics are just tics. They need to be ignored unless they're painful or disrupting a child's ability to read, or learn, or ride a bike or run. etc. etc. etc. - Meaning that they interfere with a child's life. Not our life, the child's life.

It is ok for a child (or adult) to tic you know. It's not ok for a child (or adult) to be anxious, or scared or have sleep problems... especially if the child is three years old.

Go easy on him. Maybe his transient tics were just that... transient tics. They're very common.

Unfortunately there are some very serious conditions which involve tics in young children. If you've not seen a doctor about your son's tics and his other behaviours then that's up to you, but I would personally be getting some professional advice.

I guess that's just the way I think. I guess it's just my experience.

Edited to add:

DiagnosticCriteria - DSM-IV-TR: Tourette's Disorder