Actually the LDS church temporarily abandoned polygamy around 1904 and really started enforcing this ban in 1906 with purges in 1907 that resulted in excommunication and disfellowshipment outcomes in church courts for anyone entering post-1904 polygamous relations. The LDS church issued a Manifesto in 1890 to temporarily stop new polygamous marriages in Utah so they could get statehood. During the Reed Smoot hearings it was exposed that there were still some polygamy marriage happening, including one in 1896 when Wilford Woodruff took another plural wife. Polygamous marriages were being performed on the High Seas, Mexico and out-of-state in small numbers with great secrecy. However, as far as anyone can tell, the LDS First Presidency has been very strong in enforcing the Polygamy ban since 1904.
This temporary ban is still in-force but all honest church members acknowledge that it's just a temporary ban. If Polygamy can be legalized then the LDS church will probably resume it's practice. See http://scriptures.lds.org/dc/132 for the text of the church's doctrine on polygamy.
But even if polygamy was legalized, the LDS church might not resume its practice due to political considerations. Yes they still have the 132nd Section of the D&C scripture book. But lately their prophet Gordon B. Hinckley has taken a very flexible approach on doctrine/truth. For example:
(1) Whether God was once a man and whether men can become Gods used to be a very solid teaching in the church. But now nobody really knows anything about it anymore. Its just more a couplet than anything.
(2) Valiant women are no longer required to covenant to obey the law of their husband. Now they just have to hearken to his counsel as he hearkens to the counsel of the Father.
(3) Joseph Smith and Brigham Young knew very profoundly with absolute certainty that the ancestors of the Indians were of Hebrew origin. Now with the Limited Geography Theory, this whole idea is pretty fuzzy.
(4) President Hinckley has said in the October 2001 General Conference that he doesn't know what the future holds. In the past, LDS prophets knew all sorts of things about the future. But now the prophets don't know much about the future.
(5) The church used to be very vocal about the evils of Monogamy and how the one-wife institution brought down the Roman civilization or any other civilization that practiced such an institution. However, lately, the church has not been critical of the institution of monogamy.
(6) God's law used to be DEATH ON THE SPOT for interracial marriage according to LDS prophet Brigham Young. In 1978, the LDS First Presidency announced that black men could be ordained to the Priesthood but the church leaders re-emphasized in a light manner their concerns about interracial marriage. Lately they don't talk about this much anymore.
They still believe that Brigham Young was a true prophet. Same goes for the other 13 men who are former church presidents. But they think that God has been changing his doctrines to adapt with the times through modern revelation. Just like D&C 101 was zapped in 1876 (i.e. the one that forbid polygamy) to make way for canonization of the present D&C 132, the LDS church president might get another revelation to abolish D&C 132.
I tend to think that they'll not resume practicing polygamy because overall it'll bring them more tithing and baptisms to not resume it's practice. As my sister, a returned LDS missionary, once told me, "who cares if they lie, what matters is that they save souls".
Okay, granted. Doctrine & Covenants does lay out the groundwork for the practice, based in Old Testament customs of taking multiple wives. The Manifesto issued by Wilford Woodruff in 1890 established the "restraining order" if you will, and asked that all members of the Church judge for themselves to continue the practice or be subject to reprisals per the Law of the Land. What I was getting at was that practicing members of the Church of Jesus Christ of Latter Day Saints DO NOT practice polygamy at this point in time. There are members of the Reformed Church in Southern Utah and Norther Arizona, which are pretty much the Orthodox version of the Mormons, that still practice polygamy on the sly, but if I remember right the RLDS Church frowns on the practice as well. These people are not associated with the LDS Church in any way. They chose to follow Joseph Smith's son as a prophet rather than Brigham Young.
Anyway...not to hijack this thread into a discussion of polygamy, having been raised Mormon (my wife and I attend a small Bible Church now) it still irritates me when people make false statements about the Church.