The characteristics of thunderstorms that produce terrestrial gamma-ray flashes (TGFs) observed by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) are determined using climatological and meteorological data. RHESSI observed TGFs follow diurnal, seasonal, and geographic patterns that are very similar to those of thunderstorms confirming, in part, that these events are directly connected to thunderstorm activity. The TGF producing thunderstorms are shown to be closely associated with tall (ranging from 13.6 km to 17.3 km) tropical thunderstorm systems, a finding that is consistent with theoretical expectations from models of relativistic breakdown that relate the source region to the spectral signatures observed by RHESSI. Unlike sprites, there appears to be no predilection for TGFs to occur with large thunderstorm complexes. Rather, TGF producing thunderstorms are shown to range in areal extent by several orders of magnitude. Analysis of a single TGF event within the Mozambique Channel indicates an elevated mixed phase (both liquid water and ice present) level of approximately 6 km which is consistent with the climatological findings.