The Utah case involves Kody Brown, his legal wife, Meri Brown, and three other "sister wives." It's not actually about marriage, and it doesn't challenge the right of the state to refuse to issue wedding licenses to polygamous families. The Browns are in court because they fear they will be prosecuted.
Anthony M. Kennedy offered a paean to intimate relationships defined by sexuality that easily can be transferred to the context of same-sex marriage, and potentially to polygamous marriages as well:
"The case does involve two adults who, with full and mutual consent from each other, engaged in sexual practices common to a homosexual lifestyle. The petitioners are entitled to respect for their private lives. The state cannot demean their existence or control their destiny by making their private sexual conduct a crime. Their right to liberty under the [Constitution's] due process clause gives them the full right to engage in their conduct without intervention of the government."
Kennedy emphasized in Lawrence that same-sex marriage wasn't before the court. Similarly, in an interview with the New York Times, Turley suggested that decriminalizing polygamy will not inevitably lead to a movement for polygamous marriage. But language addressed to one issue often surfaces in cases dealing with others. When Massachusetts' highest court decided to strike down the state's limitation of marriage to heterosexual couples, it cited the Lawrence opinion.
So is polygamy about to receive the same legal status that same-sex marriage now has in several states? Not in the near term. For one thing, the U.S. Supreme Court has not recognized same-sex marriage, a prerequisite, some think, for acceptance of polygamous marriage. Meanwhile, the federal court in Utah, in parallel with Lawrence, may rule simply that the Browns and other polygamous families are immune to prosecution but can't have their multiple "spiritual marriages" blessed by the law.
But, like Lawrence, a ruling sympathetic to unconventional sexual behavior could plant the seeds of a future campaign for full marriage equality. In that case, governments would have to prove that it's rational to limit marriage to two individuals, homosexual or heterosexual. That might seem obvious, but so, at one time, did the argument that marriage should be confined to opposite-sex couples.