Technically, the real answer is "neither." The Constitutional rights at issue are equal protection and due process. The Supreme Court confirmed the obvious conclusion that equal protection requires states to recognize same sex marriages. Due process does as well, although the equal protection argument is more obvious. For polyamorous relationships, the due process argument would probably be stronger than the equal protection argument.
But, I answered "both" because in layman's terms, anything Constitutional law requires is usually described as a "Constitutional right" even though that is not quite accurate. For example, people say that you have a Constitutional right to have an abortion or to have a lawyer provided for you in a criminal suit, but really those are not Constitutional rights, they're things that states need to allow/provide in order to avoid violating your Constitutional rights to privacy and due process, like the obligation to recognize same sex marriage is.