I haven't read Ziliak and McCloskey's article but it sounds like the sort of criticism Bayesians have been firing at significance testing for decades. Berger's classic book on Statistical Decision Theory has a whole list of pathologies, if you don't mind the equations. Andrew Gelman (who works in political science and stats) keeps pointing out that the null hypotheses that are set up are usually false. For example, if the null is that East Asians and Westerners do not differ on a particular performance measure, it is trivially false for all sorts of uninteresting reasons (maybe one population is a bit more motivated than the other, maybe one has more experience with video games, etc.). So if you want to find "significant" differences between cultures, in the statistical sense, all you need is a large enough sample size. Without a prior theory predicting at least the direction of the effect, for instance that East Asians should do better in a certain task, the intercultural differences we find in experiments may be entirely trivial.
Even if we had null hypotheses that made sense, and significance testing was actually what we wanted (i.e., we really wanted to control the false positive rate), it would be rendered useless by the actual practice of experimenters. One of them is to try different kinds of tests until your data come out significant. The other is to run subjects until your test comes out significant (discussed here). Finally in an issue like intercultural differences I suspect a huge file-drawer effect: no one is interested in null results. All in all the best thing to do when reading an experimental psychology paper is to ignore the p-values and look at the graphs.

Ziliak and McCloskey's book is indeed remarkable - it is also, more surprisingly, remarkably funny. In particular, the authors quite cheerfully admit that their cogent, meticulous arguments about the follies of significance testing and the cult of P will almost certainly have no effect whatsoever on current (mal)practice, especially in psychology and the social sciences. Indeed, similar points had been made by Cohen in the 60's, by Gerd Gigerenzer in the 80's, to no avail. These authors, and Ziliak and McCloskey after them, recommended paying attention to effect sizes rather than p values, reporting power, reporting confidence intervals. Yet the same emphasis on p values is characteristic of most research papers published in our fields.
If I may mention a pet peeve of mine, the practice of not just trusting p values, but actually ranking them is particularly offensive. It is rife in political science, in which one commonly finds results adorned with star-ratings (* for .05, ** for .01, *** for .001) , as if these ranks denoted different degrees of support for the hypotheses, which is of course not the case.
Another interesting aspect of this phenomenon is that almost everyone in the fields concerned (well, especially in psychology) will recognize that p values are useful but potentially misleading and that their current use as a standard of confirmation, disregarding power and effect sizes, is insane. Yet the social practice goes on! Another interesting case for the epidemiology of scientific norms.
As for the issue of cultural differences - Simon Berthelme and Olivier Morin are of course right. It is almost impossible to test two culturally different samples without finding differences (at p.05). This is like the good-bad old days of cognitive neuroscience, when all you had to do to get a study published was to plonk people in the scanner and get them to do two different things... with the guaranteed result that activations would be significantly different.
The problem for our field is not just that the null hypothesis in many cases is absurd or trivially false, it is also that "culture" is considered an independent variable in most of psychology. So when you find that e.g. Chinese and US students do not react the same way, you can publish this as evidence that "culture" matters in the domain of behavior you tested. The idea of having to explain whythis cultural difference arose or is maintained is not even considered. After all, psychologists study the effects of aging but do not think they should have to explain aging!
So it is our job to persuade everyone that "culture" is there to be explained but that in itself it explains nothing. Back to work!