We describe the results of a new reverberation mapping program focused on the nearby Seyfert galaxy NGC 3227. Photometric and spectroscopic monitoring was carried out from 2022 December to 2023 June with the Las Cumbres Observatory network of telescopes. We detected time delays in several optical broad emission lines, with H{beta} having the longest delay at {tau}_cent_=4.0_-0.9_^+0.9^ days and HeII having the shortest delay with {tau}_cent_=0.9_-0.8_^+1.1^days. We also detect velocity-resolved behavior of the H{beta} emission line, with different line-of-sight velocities corresponding to different observed time delays. Combining the integrated H{beta} time delay with the width of the variable component of the emission line and a standard scale factor suggests a black hole mass of M_BH_=1.1_-0.3_^+0.2^x10^7^M_{sun}_. Modeling of the full velocity-resolved response of the H{beta} emission line with the phenomenological code CARAMEL finds a similar mass of M_BH_=1.2_-0.7_^+1.5^x10^7^M_{sun}_ and suggests that the H{beta}-emitting broad-line region (BLR) may be represented by a biconical or flared disk structure that we are viewing at an inclination angle of {theta}i~33{deg} and with gas motions that are dominated by rotation. The new photoionization-based BLR modeling tool BELMAC finds general agreement with the observations when assuming the best-fit CARAMEL results; however, BELMAC prefers a thick-disk geometry and kinematics that are equally composed of rotation and inflow. Both codes infer a radially extended and flattened BLR that is not outflowing.