A commenter asks, "Could someone please explain how multi-[culturalism] is valuable in and of itself? I've never understood this argument, much as I've never understood why language death is such a bad thing. It seems to me that multi-[culturalism] should only be as valuable as what the component cultures bring to the table."
I certainly don't think that we should treat multiculturalism as an unalloyed good, but we should also realize that our nation was founded on multiculturalism, in two important ways. First, the premise of federalism is precisely that multiple states, which the Framers envisioned as often having substantial differences in culture, should be able to retain their cultures -- including, incidentally, the legal rules that flow from those cultures. (Within states, home rule by localities has had a similar, though lesser, mission.)
Second, the Constitution and the Bill of Rights contemplate a country with considerable range of religious views and even religious cultures. Many of the Founding-era American denominations were distinct cultural groups, such as Quakers, including groups that lived in relatively homogeneous enclaves.
These aren't just multiculturalist values. They are foundational American values. And throughout American history, they have been (or at least could be) seen as serving at least several different goals.
1. Multiculturalism as increasing minority members' happiness: Religious tolerance -- coupled with federalism and localism -- has often allowed people to live, be free, and pursue happiness in America without having to sacrifice or hide their belief systems.
2. Multiculturalism as an engine of the search for truth: Both federalism and religious diversity often produce a wide range of options -- ideological and governmental -- that then compete with each other. In federalism, this was known as the "states as laboratories of democracy" model. For religious and other ideologies, this best fits the metaphor of the "marketplace of ideas."
3. Multiculturalism as a source of valuable citizens: The tolerance for a wide range of religious belief systems has drawn more people to the nation, and has avoided their banishment. The development of the atomic bomb during World War II, which relied heavily on European (and often Jewish) scientists who fled Hitler, is one illustration of the value of ethnic and cultural tolerance; the benefits Americans have gotten from past generations of immigrants is another.
4. Multiculturalism as a source of knowledge for dealing with a multicultural world: The world is filled with lots of different cultures, whether we like it or not. Extra experience with different cultures within the U.S. helps us deal with other cultures outside the U.S. -- for instance, by giving us a pool of American citizens who actually know the foreign language or culture, or by making other citizens more familiar with dealing people of other cultures more generally.
Now it should be obvious that these are not unalloyed strengths. Multiculturalism can be a sort of domestic tension (consider the Civil War, which had cultural components, plus of course lots of other ethnically, culturally, and religiously based civil wars in other countries). Some of the cultures may teach their members to prey on outsiders (consider cultures which endorse slavery). Some of the cultures may teach their members to prey on insiders, so that tolerating the culture may give extra happiness to some members at the expense of other members. Some of the immigrants from other cultures may come to be dangers to the nation rather than assets. There are doubtless other possible problems as well.
And it should also be obvious that, because of this, we should properly calibrate our tolerance for multiculturalism with our insistence on also supporting a unified national culture. We shouldn't completely stifle all rival identities (such as Catholic, Jewish, Irish, Chinese, or whatever else), but neither should we neglect the building of an American identity. We should accommodate some religious or cultural objections to generally applicable laws, and of course we have done so for centuries in countless ways; but there are some that we shouldn't (and don't) accommodate, for instance when the objection would lead to substantial harms.
But it's also important to recognize that multiculturalism is not valueless, alien, or new. Even without reference to specific valuable aspects of specific cultures, it has some general value. It's a mistake, I think, to try to fight multiculturalism in general. Rather, we should defend those aspects of American multiculturalism that have served us well -- and are likely to continuing doing so -- and fight those aspects that are likely to be harmful.