Technically you will need a Doctors note for any medicine. Practically though I have past through the US border inspection with some OTC tablets and they did not seem interested.

Quick check on Google seems to indicate that in some states they are more controlled so it may come down to the state you enter.

Do bare in mind that some people to get addicted to codeine, as it is metabolized in the body into morphine (hence the restrictions). If they are prescription strength and you really need them best get a doctors certificate.