We have examined a complete sample of 1 s resolution Vela 5B data for all low-mass X-ray binaries for which they can be extracted without source confusion, and searched for periodicities of more than 1 day using Fourier analysis and period folding techniques. We confirm the existence of a 174.6 day period in X1820 - 303 with extremely high confidence, derive an accurate ephemeris for the modulation, and model the form of the folded light curve. We have searched the data from X1916 - 053 for the published 199 day period; no significant periodicities are detected in the raw data, although the candidate period appears in power spectra constructed from a filtered data set. We conclude that this period detection is not firm, with a false alarm probability (FAP) in the range 10% - 20%. For Cygnus X-2 we find a significant periodicity of 77.34 days with a false alarm probability of less than 10(-5); this may be associated with an accretion disk precession time scale. We detect the 4.8 hr orbital period of Cygnus X-3, and show in addition that the power spectrum shows variability on time scales up to 200 days. No periodicities greater than 1 day with a significance greater than 2-sigma were found in the following sources: NGC 1851, X0614 + 091, X0918 - 549, X0921 - 630, X1254 - 690, X1627 - 673, X1636 - 536, X1735 - 444, X1822 - 371, X1822 - 000, Serpens X-1, X1957 + 115, and M15. For these sources we have placed upper limits on the existence of periods of order hundreds of days, and also upon the detection of some periods in the literature of approximately a fev; days. Our analysis suggests that while long-term cyclic variability is common among high-mass binaries, it is a rare occurrence in the low-mass systems.