A Fourier analysis of the magnitudes and timing of the Phanerozoic mass extinctions (MEs) demonstrates that many of the periodicities claimed in other analyses are not statistically significant. Moreover we show that the periodicities associated with oscillations of the Solar System about the galactic plane are too irregular to give narrow peaks in the Fourier periodograms. This leads us to conclude that, apart from possibly a small number of major events, astronomical causes for MEs can largely be ruled out.