My understanding is that multiple marriages (i.e. having more than one state issued marriage license) is not legal in Utah. However, it is now legal in Utah to "cohabit" or live with one more than one woman ("wife") as in the practice of polygamy as part of one's religion. See excerpt below.
Utah, I think, had previously been after the Browns for cohabitating as one big family (hence the move to Nevada), but since the decision excerpted below, they will not now attempt to prosecute them and they could return and live in Utah if they so desired.
However, Kody still could not go to Utah and legally marry all of his "wives." I believe Meri is still his only legal wife.
Excerpt from New York Times 12/14/2013:
Judge Clark Waddoups of United States District Court in Utah ruled late Friday that part of the state’s law prohibiting “cohabitation” — the language used in the law to restrict polygamous relationships — violates the First Amendment guarantee of free exercise of religion, as well as constitutional due process.
He left standing the state’s ability to prohibit multiple marriages “in the literal sense” of having two or more valid marriage licenses.