The same reasoning that finds a right to same-sex marriage in the U.S. Constitution also supports the right of more than two people to marry, In other words, polyandry, polygyny, and polyamory, known collectively as polygamy. For that matter, dropping the ban on incest is likely as well.
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I can imagine states where most residents do not concur with today's decision at least considering getting out of marriage licensure entirely. Perhaps in those jurisdictions "marriage" would become a contractual arrangement entered into by simple signature before witnesses.
Any associated religious ceremony would require mutual agreement of the contracting parties plus the church in question, and have no legal standing. If a church found certain marital configurations doctrinally offensive, they would not be required to bless same. Presumably other churches would discover they had flexibility in such matters and be available.
Smart people would include in their contract those provisions which would apply in the event of a dissolution or other non-performance, including penalties and provisions for children, pets, chattel and real property. "Divorces" might be concluded by a mediator, arbitrator, or civil court judge as specified in the contract.
Under these circumstances, a "marriage" would include whatever lawful features upon which the contracting parties could agree, and only those. Think of the employment for attorneys such an environment would provide - a mostly new branch of legal practice related to the prior practice of drafting prenuptial agreements.
Pressing individuals to make explicit their expectations of each other before marriage might be a healthy step toward reducing divorce.
