See http://www.religioustolerance.org/lds_poly.htm
In actions reminiscent of the "same-sex marriage" debate which started in the late 1990's, the Mormon practice of polygyny was countered by the federal Morrill Act of 1862 and the Edmunds Act of 1882, both of which prohibited the practice within the U.S. The Church claimed that the federal government had no jurisdiction to regulate marriage and other internal church practices. They also claimed that the act was a violation of Mromon's First Amendment rights. In 1879, the US Supreme Court declared that the Morrill Act was constitutional, that the government had a right to enforce marital standards, and that polygyny was a barbarous practice.
The federal government canceled the citizenship rights of polygamous Mormons. They were no longer allowed to vote, run for public office, or serve on a jury. Eventually, the government disincorporated the Church and began to confiscate its assets. In 1890, the Supreme Court determined that the government could take away citizenships from all members of the church.
I doubt that the above is "good" law these days. :)