Oscillations in solar coronal loops appear to be a common phenomenon. Transverse and longitudinal oscillations have been observed with both the Transition Region and Coronal Explorer and Extreme Ultraviolet Imaging Telescope imaging experiments. Damped Doppler shift oscillations have been observed in emission lines from ions formed at flare temperatures with the Solar Ultraviolet Measurements of Emitted Radiation Spectrometer. These observations provide valuable diagnostic information on coronal conditions and may help refine our understanding of coronal heating mechanisms. I have initiated a study of the time dependence of Doppler shifts measured during flares with the Bragg Crystal Spectrometer (BCS) on Yohkoh. This Letter reports the detection of oscillatory behavior in Doppler shifts measured as a function of time in the emission lines of S XV and Ca XIX. For some flares, both lines exhibit damped Doppler shift oscillations with amplitudes of a few km/s and periods and decay times of a few minutes. The observations appear to be consistent with transverse oscillations. Because the BCS observed continuously for almost an entire solar cycle, it provides numerous flare data sets, which should permit an excellent characterization of the average properties of the oscillations.