I felt like weighing in on this issue after seeing the new campaign and site by the National Organization for Marriage in the US.
The lobbies and congregations that make up the Christian Right have realised that those who are less religious than they are (i.e. the majority) will not accept the dogmatic arguments from Scripture with which they have convinced themselves, and are using broader approaches. This is reasonable, and is the best way for religious organisations to pursue their interests in a secular society, if the replacement arguments are actually valid. If not, it's a form of deception.
I'm about to summarise the non-Scriptural arguments against same-sex marriage (SSM) by paraphrasing the site above. I'm doing my best not to create any straw men with this approach, but if I do so anyway, tell me off.
1. Same-sex marriages deprive children of either a mother or a father.
This is true, but that mother or father is replaced with either another father or another mother. In principle, the number of adults caring for the children is the same, and the proportion of men and women raising the children depends very little on the parents themselves. Children without mothers for example can have aunts, grandmothers, big sisters, cousins, nannies, friends of the parents and so on.
In practice, no significant difference in development, social life or even sexual tendency has been found between children with same-sex parents and children with different-sex parents. Anti-SSM literature appears to focus entirely on studies of children of single parents, who are missing a mother or father for very different reasons. Such research, while important, is irrelevant to the issue of the gender of existing parents.
2. Public and legal acceptance of same-sex marriage will reduce religious freedom. Believers, churches and religious charities such as the Salvation Army will be unable to practice unless they endorse same-sex marriage.
Individual religious freedom and that of churches will be unaffected. It's already illegal in the USA to discriminate against homosexuals, but the right of evangelical Christians and their pastors to believe, announce and advertise that homosexuality is sinful is protected by free speech and, importantly, "freedom of religion".
What will be curtailed is individual freedom to discriminate in practical ways, as has already happened with progress in racial equality and gay rights. The central example is the staff at artificial insemination clinics and adoption agencies: some of them don't want to be forced to give kids to gay couples. If their reasons for this are religious, their faith is about to conflict with their current jobs, but they are free to find work elsewhere in their respective industries. If instead their reason actually is reason 1 above, it's not a good reason.
Finally, religious charities and other organisations have nothing additional to worry about. They're already in trouble if they discriminate against gays. I don't see why they would discriminate against children of gay couples, because
a. the kids' upbringing isn't the kids' fault, and
b. if they really think same-sex parents are worse, they would conclude that the kids need more help.
3. If we change the definition of marriage, what's to stop us from changing it further to allow polygamy, marriage to animals, underage marriages, etcetera? (Paraphrased from a point on the site's .pdf handout, Why Marriage Matters. To be fair, these guys only mentioned polygamy.)
There is indeed an extremely small minority which would like marriage to be further expanded in these ways. Some, like those in the Fundamentalist Church of Jesus Christ of Latter-Day Saints, go ahead and practice polygamy without looking for endorsement. Many others have less formal "open marriages".
The difference is in the practical benefits of each change. Once same-sex marriage is allowed, every adult will be allowed to marry a consenting adult of their choice, with whom they can have a happy intimate relationship, and raise a family in accordance with their common human desires. This gives everyone an ability that was once only available to some, and so negates a now-arbitrary piece of discrimination.
Other changes to marriage do not confer similar benefits, and carry additional drawbacks. Polygamy does not extend the chance for marriage and a family to anyone who doesn't already have it. Underage marriage and marriage to animals are cruel to the partner who is unable to consent.
This is why it most benefits humanity to extend marriage so far and no further, and why no one need be afraid that the floodgates will open, so to speak.
Since it does appear that the secular arguments presented by the anti-SSM movement have little value, the only reasons left are the Scriptural ones they are so eager to keep in the background. Once those are the remaining line of defense, they have no place in the political sphere, at least in a country which has declared church and state separate. For the rest of the world, though, it's a bit muckier.