It has come out in support of certain rights for same-sex couples:
The Church does not object to rights (already established in California) regarding hospitalization and medical care, fair housing and employment rights, or probate rights, so long as these do not infringe on the integrity of the family or the constitutional rights of churches and their adherents to administer and practice their religion free from government interference. - The Divine Institution of Marriage
That's great, but it comes with the "so long as" qualifier (part of which [infringement of constitutional rights of churches and adherents] is totally understandable and probably necessary, the other part of which [infringing on the 'integrity of the family'] is completely subjective and allows for plenty of wiggle room should the church need it) and says nothing of civil unions, adoption rights, taxation, insurance, guardianship...
It has also clearly and officially implied it does not support giving same-sex couples all of the same rights and responsibilities as mixed-sex couples:
Legalizing same-sex marriage will affect a wide spectrum of government activities and policies. Once a state government declares that same-sex unions are a civil right, those governments almost certainly will enforce a wide variety of other policies intended to ensure that there is no discrimination against same-sex couples. This may well place "church and state on a collision course." - The Divine Institution of Marriage (emphasis added)
Classic "us or them". Of course, they expound on other reasons to bolster their opposition to same-sex marriage, reasons I believe are completely debatable if not debunkable. But my point here is that, to the church, this apparently isn't just about the word 'marriage', folks. So at the risk of spurring some of you to change your minds in a direction I disagree with, I say to my LDS-faithful friends and family: be careful about where you draw your lines, or you may find yourself unexpectedly at odds with 'the brethren' in your attempts to be 'tolerant'.