This is a civil matter, so while they should refund your money immediately, you may have to pull some teeth to get them to act. (sorry, couldn't help it)
Send a strong letter requesting payment by a certain date. Something like, "According to ABC insurance, you received payment from them two months ago. I contacted you three times already regarding reimbursement to no avail. I hereby demand reimbursement in the amount of $1000 by Friday, December 19th or I will be forced to take further action."
Might even want to send it certified mail, return receipt requested. That should do it -- a threat of legal action without explicitly saying so should get the point across. I'd expect a check pretty quickly. If it were me and they still don't pay by the deadline, I'd take them to small claims court (no lawyer needed). You won't get interest or penalties, but the costs of small claims court will be included in the judgment. I doubt it goes that far.
Oh, and use Yelp to find another dentist!