First off, I am LDS/Mormon. I read an article about this very topic a month or so ago. I'm trying to find it. I'm almost certain I saved it so it's just a matter of trying to find it. Until then, let me summarize what it said.
Back during the 1890s when polygamy was outlawed, it was the republican party heading the opposition. The LDS population almost exclusively voted for Democrat candidates in response. The Church leaders did not want the church to be thought of as a one party group and sent out one the the leaders, I believe a member of the Quorum of the Seventy, to speak to the members about this matter. The basics of his message is that it is OK for a good member to vote republican. What should be important is to vote your conscious and to choose a candidate that best represents your values. His message was well receieved. As a rule, when the leaders speak, the members follow without hesitation. The shift has continued until today the Church is heavily republican, or at least thought to be. I agree with what an earlier poster put up about members outside of Utah being much more liberal. I have lived in Los Angeles and Austin, and members in both towns are fairly liberal. Not spoken of in church, but it is the private beliefs, and that is more important. You can see there are members elected to congress who aren't republican, including the highest ranked member of the Church.
http://www.reviewjournal.com/lvrj_home/2002/Sep-08-Sun-2002/news/19525007.htmlThe Church today doesn't seem to interested in being seen as a one party group. Personally, I feel the reason behind that is that the Church tends to operate more as a business, and has started to act like one. Also, with the emphasis on education, the members have moved upward in income as well. With that comes the idea one should vote republican to protect that money. We all know thats not correct, or at the very least not all of the story. However, it is a common perception.
I have lived in SLC as well and the Church is different in Utah. It is like living in a bubble, very sheltered, with members more interested in following what the leaders say or what they think the leaders would prefer, rather than think an issue out for themselves. I don't want to stereotype everyone, but that is my impession from living there.
I've rambled long enough. I'll try to find the link I talked about earlier.