The EUV Imaging Spectrometer (EIS) on the Hinode satellite is capable of measuring emission line center positions for Gaussian line profiles to a fraction of a spectral pixel, resulting in relative solar Doppler-shift measurements with an accuracy of less than a km/s for strong lines. We show an example of the application of that capability to an active region sit-and-stare observation in which the EIS slit is placed at one location on the Sun and many exposures are taken while the spacecraft tracking keeps the same solar location within the slit. For the active region examined (NOAA 10930), we find that significant intensity and Doppler-shift fluctuations as a function of time are present at a number of locations. These fluctuations appear to be similar to those observed in high-temperature emission lines with other space-borne spectroscopic instruments. With its increased sensitivity over earlier spectrometers and its ability to image many emission lines simultaneously, EIS should provide significant new constraints on Doppler-shift oscillations in the corona.