The Fermi Gamma-ray Burst Monitor (GBM) is an all-sky gamma-ray monitor well known in the gamma-ray burst (GRB) community. Although GBM excels in detecting the hard, bright extragalactic GRBs, its sensitivity above 8 keV and its all-sky view make it an excellent instrument for the detection of rare, short-lived Galactic transients. In 2010 March, we initiated a systematic search for transients using GBM data. We conclude this phase of the search by presenting a three-year catalog of 1084 X-ray bursts. Using spectral analysis, location, and spatial distributions we classified the 1084 events into 752 thermonuclear X-ray bursts, 267 transient events from accretion flares and X-ray pulses, and 65 untriggered gamma-ray bursts. All thermonuclear bursts have peak blackbody temperatures broadly consistent with photospheric radius expansion (PRE) bursts. We find an average rate of 1.4 PRE bursts per day, integrated over all Galactic bursters within about 10 kpc. These include 33 and 10 bursts from the ultra-compact X-ray binaries 4U 0614+09 and 2S 0918-549, respectively. We discuss these recurrence times and estimate the total mass ejected by PRE bursts in our Galaxy.