How important is it to have publications to get a math postdoc?
[The following answer is from the point of view of a tenured professor of theoretical mathematics at an American university.  I believe that most of what I say applies in many places outside of the United States, but not everywhere, and I will not try to say exactly where I think it applies.]

A generation ago the publication culture within mathematics was quite different from other STEM fields.  Students leaving their PhDs were not expected to have any publications at all, and I believe the majority of them did not.  This phenomenon was pretty extreme, to the extent that for many eminent mathematicians you can see that their first publication is work done at an REU when they were an undergraduate, often followed by a gap of about five years, then their "real" publications begin only after their thesis.  A postdoctoral job was awarded mostly based on the relatively brief description of the thesis work provided by the student and (more importantly, I think) the student's thesis advisor.

[In my case, I graduated with a math PhD from Harvard in 2003 and my first paper wasn't until 2005.  To make sure I wasn't overextrapolating from my own experience, I went back to look at my classmates at Harvard.  I found a few cases in which a paper from their thesis appeared slightly before they graduated, but in most cases their first paper appears 1-2 years after graduation.]

Nowadays, a few graduating PhDs can still function on the above model, but only under really ideal conditions: top program, advisor with enormous pull who says great things about you in the letter.  To get personal again:

1) I am about to graduate my fifth PhD student at UGA.  All of these students had at least a submitted paper by the time they graduate; in most cases they had one or more accepted papers.

2) Part of the application for postdoc positions at UGA [of which there are several kinds, but this is a common feature] is a publication list.  I have been responsible for making offers to postdocs for a while now, and I am struggling to remember making an offer to someone who didn't at least have a publicly available submitted preprint.  Sometimes we have had candidates who were otherwise of interest but it's hard to pull the trigger based on seeing none of their work when there are so many other applicants who have multiple papers.

And to get less personal:

3) Whereas 15 years ago good students from top departments didn't need and most often didn't have any preprints, nowadays it is more common for good students from top departments to exit with several papers.

I do however want to add an important caveat: more mathematics done is better than less mathematics done.  [As is not so surprising!]  However, more papers are not better than fewer papers if the more papers don't create the impression of having done more work of significance.  In particular, the cultural of theoretical mathematics very much works against the LPU model espoused in another answer: writing too many papers on the same topic "without new ideas" creates a poor impression.  If two mathematical journals differ by one tier, than having one paper in the better journal is better than having at least two papers in the worse journal.  If the journals differ by more than one tier, having one paper in the better journal is probably better than having any number of papers in the worse journal.  The top journals in mathematics want to publish important, substantial, difficult, breakthrough work: if you split one such 40 page paper up into four ten page papers, then you have four papers that each do very partial things and are not going to be published in nearly as good venues.  Moreover you will get the reputation as having "more papers than theorems," which is not good.  

I would say that the following is a good publication model for a new math PhD: have a portion of your thesis already submitted to a good journal and have one other reasonable paper [possibly on a different topic] published elsewhere.  More papers than that is not necessarily helpful: the quality of your thesis work still matters more.