No, there doesn't appear to be very good scientific support for this popular idea.
This 2011 article at ScienceBasedMedicine appears to cover all the important bases (the article is well-referenced, I'll reproduce a few key links here):
Synchrony Is difficult to define
Consider that the normal menstrual cycle can vary from 21 to 35 days
and can last 2 to 7 days. Consider that some women are regular and
consistent, while others have variable patterns, even “regularly
irregular” patterns. Consider that anovulatory cycles and other
conditions often lead to menstrual irregularities that fall outside
the normal range. Consider that strenuous exercise and other life
events can affect menstruation. Put all that together, and you can see
that often cycles will overlap simply by chance, and that it is
difficult to define synchrony.
Research is mixed at best
A Scientific American article did a good job of reviewing the literature as of 2007.
Suffice it to say that about half the published papers support the
synchronization hypothesis and half don’t; and the half that do have
been harshly criticized for their poor design and poor statistical
analyses. So we haven’t reached a consensus, but it’s looking more
likely that synchronization is a myth.
Comparative studies on animals have been negative
It was originally thought that other primates and other mammals
demonstrated menstrual synchrony due to pheromones, but recent studies
have shown that it doesn’t occur in chimpanzees, hamsters, mandrills,
or golden tamarins.
The proposed mechanism (human pheromones) has little to no scientific support
In humans, pheromones have been postulated and even sold as sexual
attractants. But there is little or no peer-reviewed evidence to
suggest that any pheromone influences human behavior. No human
pheromones have been identified, and the vomeronasal organ that
detects pheromones in other mammals is rudimentary and nonfunctional
The first comment on the article, by "Al Morrison", summarizes it nicely:
No agreed upon definition of menstrual synchronization + no plausible
mechanism + poor research supporting the hypothesis + split research
results + no support from comparative research + a tendency to over
report synchronization = MYTH