Measuring the masses of supermassive black holes in active galactic nuclei (AGN) allows us to trace their evolution over cosmic time and understand how black holes coevolve with their host galaxies. We present a new technique to measure black hole masses and constrain the structure of the broad line region in AGN using reverberation mapping data. We begin by developing a simply parameterized phenomenological model of the broad line region geometry and dynamics and apply this model to high-quality reverberation mapping data for six AGN from the Lick AGN Monitoring Project 2008 and 2011 datasets. The results of this analysis provide the most precise AGN black hole masses from reverberation mapping to date and the first detailed constraints on the geometry and dynamics of the broad line region emission. Specifically, we find that the shape of the broad line region is generally a close to face-on thick disk with preferential emission from the far side, and that the dynamics range from inflow to near-circular orbits. In addition, we present photometric AGN light curves using image subtraction for the Lick AGN Monitoring Project 2011 dataset as a first step towards modeling the broad line region in a larger sample of AGN.