In an excellent recent post about why religions often have different criteria for membership than the larger society might prefer, Megan McArdle wrote:

There isn't anyone with the authority, as far as I know, to tell you you aren't a member of one of those faiths. You can be expelled from a congregation, to be sure, but no minister can declare that you are not a Presbyterian, no rabbi strip your Jewishness from you. When they hear a bishop telling someone to stop calling themselves Catholic, it sounds like that bishop telling that person to stop proclaiming their beliefs.
It's a little more complicated than that, however. For one thing, unlike most mainline protestant denominations, or any Jewish ones, Catholics are pretty firm on what beliefs you have to embrace to be a member. [Aren't the Orthodox Jews pretty strict? -- ed. Yes, but as far as I know, technically, you don't have to believe in anything they say, as long as you obey the Law.]

Megan is almost completely mistaken about Judaism, especially the Orthodox variety. Yes, we are pretty strict about our requirements.
First, it is true that Judaism views membership as an entrance into a roach motel: you can check in, but you can't check out. (This is, by the way, one reason why Jews historically discourage prospective converts before even beginning to consider them; with respect to those who aren't going to be able or willing to fulfill all their new requirements, we're doing them a favor.) That being said, though, there are two main problems with Megan's points:
1) First, Judaism does recognize that you can do certain things that put you outside the pale and cause you to be considered, in many respects, as being outside the Jewish community. (One example of such exclusion is that such a person, if male, would not be allowed to count towards the quorum required for a public prayer service ("minyan").) What do you have to do in order to achieve this feat? The most common example in the halakhic sources is "violating the Sabbath in public," but public professions of heresy will do nicely as well. For a decent overview of heresy in Judaism, click here and scroll down. From the same source, about which I don't know much, click here for an overview of Jewish priciples of faith.

(Digression #1: Nowadays, do Orthodox Jews apply the status of "public Sabbath violator," with the accompanying communal exclusions, to everyone who doesn't observe the Sabbath in accordance with Orthodox standards - i.e., the overwhelming majority of Jews today? No, for a variety of reasons beyond the scope of this discussion - we can take it up in the comments. The point for these purposes is that there is something you can do that puts you outside the Jewish communal pale, even if it technically doesn't remove your Jewishness. End of digression.)

2) In general, Judaism does in fact require you to believe certain things. If you missed the link, click here for an overview.
Nevertheless, Megan's misperception is pretty common. Where does it come from? I can think of a couple of reasons:
A) The closest source for the anti-dogmatic view cited by Megan is the work of Moses Mendelssohn. I am not an expert in Mendelssohn's work, but I do know that his views have never gained wide acceptance within Orthodox thought. It is thus misleading to say that, as far as the Orthodox are concerned, Mendelssohn's views on dogma represent "Judaism."
B) Codifications of Jewish dogma (most notably, this one) only became popular in medieval times, as a response to widespread Christian (and Muslim, I think) works of dogma. Those codifications were usually seen as worthy efforts, but were outside the mainstream works of halakhic scholarship - where the intellectual action has been for over 2,000 years. Those codifications' impact on halakhic practice was relatively minimal, and while Maimonides' "13 principles" have become the most commonly accepted codification of Jewish dogma, even those principles have not been universally accepted within Orthodox Judaism.
C) Most importantly, the extreme non-dogmatic view is built on a certain truth, as hinted at above: most of the intellectual energy of Orthodox Judaism is dedicated toward analysis of the system and details of halakha, rather than the fine points of belief. But that doesn't mean that belief is optional; the emphasis instead reflects the following points:
(i) Why, exactly, would you buy into the halakhic system in the first place if you don't believe that it reflects the will of God?
(ii) There is a famous Talmudic statement that says, essentially: "Even if you are not performing [an obligation] for the right reasons, perform it anyway, beacuse one who does so will eventually come to do so for the right reasons." ("Mi-she-lo lishma, ba lishma.") That statement reflects several truths as a guidline for dealing with issues such as a temporary faith crisis, but it is far from a sanction for dispensing with faith altogether.

Debate invited...