We have started a long-term reverberation mapping (RM) project using the Wyoming Infrared Observatory 2.3 m telescope titled "Monitoring AGNs with H beta Asymmetry" (MAHA). The motivations of the project are to explore the geometry and kinematics of the gas responsible for complex H beta emission-line profiles, ideally leading to an understanding of the structures and origins of the broad-line region (BLR). Furthermore, such a project provides the opportunity to search for evidence of close binary supermassive black holes. We describe MAHA and report initial results from our first campaign, from 2016 December to 2017 May, highlighting velocity-resolved time lags for four active galactic nuclei (AGNs) with asymmetric H beta lines. We find that 3C 120, Ark 120, and Mrk 6 display complex features different from the simple signatures expected for pure outflow, inflow, or a Keplerian disk. While three of the objects have been previously reverberation mapped, including velocity-resolved time lags in the cases of 3C 120 and Mrk 6, we report a time lag and corresponding black hole mass measurement for SBS 1518+593 for the first time. Furthermore, SBS 1518+593, the least asymmetric of the four, does show velocity-resolved time lags characteristic of a Keplerian disk or virialized motion more generally. Also, the velocity-resolved time lags of 3C 120 have significantly changed since previously observed, indicating an evolution of its BLR structure. Future analyses of the data for these objects and others in MAHA will explore the full diversity of H beta lines and the physics of AGN BLRs.