For regular firearms you only have to comply with the laws of the state to which you are moving. Since Utah (presently) has no state-specific restrictions on ownership or possession of AR-15 rifles that means in this case, as you say, "no problem."
(The only exception would be registered NFA items – e.g., machine-guns, suppressors, SBR, SBS, DD, AOW – for which the BATFE typically requires notice when you are moving them interstate. But if you have gone through the trouble of registering such an item, you probably already know that.)
Finally, you have to exercise some care in the actual interstate transportation of firearms: If you are passing through a jurisdiction where possession of them is restricted then you have to follow the rules provided in the Firearms Owners Protection Act. In particular:
Under FOPA, notwithstanding any state or local law, a person is
entitled to transport a firearm from any place where he or she may
lawfully possess and carry such firearm to any other place where he or
she may lawfully possess and carry it, if the firearm is unloaded and
locked out of reach. In vehicles without a trunk, the unloaded firearm
must be in a locked container other than the glove compartment or
console. Ammunition that is either locked out of reach in the trunk or
in a locked container other than the glove compartment or console is
also covered.
