The Large Area Telescope (LAT) aboard the Fermi Gamma-ray Space Telescope provides an unprecedented opportunity to study gamma-ray blazars. To capitalize on this opportunity, beginning in late 2007, about a year before the start of LAT science operations, we began a large-scale, fast-cadence 15 GHz radio monitoring program with the 40 m telescope at the Owens Valley Radio Observatory. This program began with the 1158 northern (δ > –20°) sources from the Candidate Gamma-ray Blazar Survey and now encompasses over 1500 sources, each observed twice per week with about 4 mJy (minimum) and 3% (typical) uncertainty. Here, we describe this monitoring program and our methods, and present radio light curves from the first two years (2008 and 2009). As a first application, we combine these data with a novel measure of light curve variability amplitude, the intrinsic modulation index, through a likelihood analysis to examine the variability properties of subpopulations of our sample. We demonstrate that, with high significance (6σ), gamma-ray-loud blazars detected by the LAT during its first 11 months of operation vary with almost a factor of two greater amplitude than do the gamma-ray-quiet blazars in our sample. We also find a significant (3σ) difference between variability amplitude in BL Lacertae objects and flat-spectrum radio quasars (FSRQs), with the former exhibiting larger variability amplitudes. Finally, low-redshift (z < 1) FSRQs are found to vary more strongly than high-redshift FSRQs, with 3σ significance. These findings represent an important step toward understanding why some blazars emit gamma-rays while others, with apparently similar properties, remain silent.