We present results for the first three years of OzDES, a six year programme to obtain redshifts for objects in the Dark Energy Survey (DES) supernova fields using the 2dF fibre positioner and AAOmega spectrograph on the Anglo-Australian Telescope. OzDES is a multi-object spectroscopic survey targeting multiple types of targets at multiple epochs over a multiyear baseline and is one of the first multi-object spectroscopic surveys to dynamically include transients into the target list soon after their discovery. At the end of three years, OzDES has spectroscopically confirmed almost 100 supernovae, and has measured redshifts for 17 000 objects, including the redshifts of 2566 supernova hosts. We examine how our ability to measure redshifts for targets of various types depends on signal-to-noise ratio (S/N), magnitude and exposure time, finding that our redshift success rate increases significantly at a S/N of 2-3 per 1-angstrom bin. We also find that the change in S/N with exposure time closely matches the Poisson limit for stacked exposures as long as 10 h. We use these results to predict the redshift yield of the full OzDES survey, as well as the potential yields of future surveys on other facilities such as (i.e. the 4-m Multi-Object Spectroscopic Telescope, the Subaru Prime Focus Spectrograph and the Maunakea Spectroscopic Explorer). This work marks the first OzDES data release, comprising 14 693 redshifts. OzDES is on target to obtain over 30 000 redshifts over the 6-yr duration of the survey, including a yield of approximately 5700 supernova host-galaxy redshifts.