This work presents a single-structure 3-axis Lorentz force magnetometer (LFM) based on an AlN-on-Si MEMS resonator. The operation of the proposed LFM relies on the flexible manipulation of applied excitation currents in different directions and frequencies, enabling the effective actuation of two mechanical vibration modes in a single device for magnetic field measurements in three axes. Specifically, the excited out-of-plane drum-like mode at 277 kHz is used for measuring the x- and y-axis magnetic fields, while the in-plane square-extensional mode at 5.4 MHz is used for measuring the z-axis magnetic field. The different configurations of applied excitation currents ensure good cross-interference immunity among the three axes. Compared to conventional capacitive LFMs, the proposed piezoelectric LFM utilizes strong electromechanical coupling from the AlN layer, which allows it to operate at ambient pressure with a high sensitivity. To understand and analyze the measured results, a novel equivalent circuit model for the proposed LFM is also reported in this work, which serves to separate the effect of Lorentz force from the unwanted capacitive feedthrough. The demonstrated 3-axis LFM exhibits measured magnetic responsivities of 1.74 ppm/mT, 1.83 ppm/mT and 6.75 ppm/mT in the x-, y- and z-axes, respectively, which are comparable to their capacitive counterparts.