A search has been made for sinusoidal periodic variations in the $^{8}\mathrm{B}$ solar neutrino flux using data collected by the Sudbury Neutrino Observatory over a 4-year time interval. The variation at a period of 1 yr is consistent with modulation of the $^{8}\mathrm{B}$ neutrino flux by the Earth's orbital eccentricity. No significant sinusoidal periodicities are found with periods between 1 d and 10 years with either an unbinned maximum likelihood analysis or a Lomb-Scargle periodogram analysis. The data are inconsistent with the hypothesis that the results of the recent analysis by Sturrock et al., based on elastic scattering events in Super-Kamiokande, can be attributed to a $7%$ sinusoidal modulation of the total $^{8}\mathrm{B}$ neutrino flux.