We present the results of a program to monitor the structure of the radio emission in 42 γ-ray bright blazars (31 quasars and 11 BL Lac objects) with the Very Long Baseline Array (VLBA) at 43, 22, and occasionally 15 and 8.4 GHz, over the period from 1993 November to 1997 July. We determine proper motions in 33 sources and find that the apparent superluminal motions in γ-ray sources are much faster than for the general population of bright compact radio sources. This follows the strong dependence of the γ-ray flux on the level of relativistic beaming for both external radiation Compton and synchrotron self-Compton emission. There is a positive correlation (correlation coefficient r = 0.45) between the flux density of the VLBI core and the γ-ray flux and a moderate correlation (partial correlation coefficient r = 0.31) between γ-ray apparent luminosity and superluminal velocities of jet components, as expected if the γ-ray emission originates in a very compact region of the relativistic jet and is highly beamed. In 43% of the sources the jet bends by more than 20° on parsec scales, which is consistent with amplification by projection effects of modest actual changes in position angle. In 27 of the sources in the sample there is at least one non-core component that appears to be stationary during our observations. Different characteristics of stationary features close to and farther from the core lead us to suggest two different classes of stationary components: those within about 2 mas of the core, probably associated with standing hydrodynamical compressions, and those farther down the jet, which tend to be associated with bends in the jet.