Eugene Volokh of The Volokh Conspiracy, one of the blogosphere's most overrated blogs, sparks a discussion on moral relativism and liberalism with this post. When reading this discussion, the first thing to do is ignore the philosophical points. Volokh's philosophical statements are interesting and true, but ultimately irrelevant, for the reasons that I will discuss in a moment, and Matthew Yglesias' display an ignorance of ethical theories that one would not expect from someone who recently received a degree in philosophy from Harvard, so there's really no point in paying attention to this part of the debate (which means you can ignore entirely the contribution at Crooked Timber, which is almost completely devoted to Yglesias' dilettantish philosophical digression). Instead, as Kevin Drum notes, the real issue is the "common sense" view of moral relativism, and why so many conservatives (and even some liberals) associated it with liberalsm.
So what is the common sense concept of "moral relativism?" Most believe moral relativism to be the position that all moral values are equally valuable, or that all ethical views are equally valuable. The type of relativism most likely to be discussed by non-philosophers is cultural relativism, in which (according to the common sense view of moral relativism) the ethical systems of all cultures are equally valuable. The question raised by Volokh's post is whether these labels apply to liberals. The answer, of course, is that these labels (with their "common sense" meanings) don't really apply to anyone. There is no one who believes, and certainly no one who acts on the belief, that all moral values are equally valuable. That's just silly, and probably humanly impossible. It is true, as Volokh says, that both liberals and conservatives are always situationalists to some degree (though often in the guise of a deontological ethics where the morally-relevant concepts, e.g. the concept of murder, are defined in such a way that they mask the situationalist aspects the ethical system), but situationalists are not necessarily relativists. Liberals aren't cultural relativists, either. While cultural relativism is very popular in anthropology, even there it does not hold that all ethical systems are equally valuable. Instead, it is a methodological stance that involves approaching all cultures from an objective, non-evaluative standpoint when describing them scientifically. The truth is that in many cases, liberals are as far from moral relativism, and cultural relativism in particular, as one can be, believing as they do in things like universal human rights, and universal concepts of cruelty (a point that Volokh notes in his original post, but which isn't mentioned in any of the responses).
So why are liberals shackled with the "moral relativist" label? I think there are several reasons. The primary reason, however, is framing. Conservatives believe in strict moral principles (though they don't always act on those beliefs). One way to villify their opponents is to cast them as people who are morally corrupt, and from the conservative perspective, one way to be morally corrupt is not to recognize the conservative moral system as the one true, empirically verifiable, absolute, and perhaps divinely inspired moral system. This also explains the view of liberals as atheists (rather than the other way around, as Yglesias suggests). While the vast majority of liberals in the United States are Christians, it is rhetorically expedient for conservatives to frame them as secularists, or even atheists, because it implies a lack of moral integrity. If the one true and absolute moral system is divinely inspired, and you don't believe in the divine, how can you be truly moral? You can't, of course.
While it should be obvious that liberals are not "common sense" moral relativists, liberals rarely do anything to counter the perception that they are are. Tolerance is a concept central to contemporary liberalism, but liberals rarely express it very well. Tolerance is never the view that all cultural practices should be treated as equal. Liberals would not be liberals if they believed that the exploitation and marginalization of women in Arab cultures, or other forms of cruelty justified through religious or cultural institutions, are moral. However, this is the view that conservatives attribute to them, and one rarely hears liberals attempting to debunk this misrepresentation. This is yet another example of conservatives being master framers, and liberals being lost in the rhetorical sea. In an almost ironic sense, then, it's the conservatives who've embraced the world-view most like relativism (see here), and have used it to frame liberals as relativists. Liberals, because of their more anti-relativistic world-view, have had a hard time countering the conservative framing.