A black hole (BH) traveling through a uniform, gaseous medium is described by Bondi–Hoyle–Lyttleton (BHL) accretion. If the medium is magnetized, then the black hole can produce relativistic outflows. We performed the first 3D, general-relativistic magnetohydrodynamic simulations of BHL accretion onto rapidly rotating black holes using the H-AMR code, where we mainly varied the strength of a background magnetic field that threads the medium. We found that the ensuing accretion continuously drags the magnetic flux to the BH, which accumulates near the event horizon until it becomes dynamically important. Depending on the strength of the background magnetic field, the BHs can sometimes launch relativistic jets with high enough power to drill out of the inner accretion flow, become bent by the headwind, and escape to large distances. For stronger background magnetic fields, the jets are continuously powered, while at weaker field strengths they are intermittent, turning on and off depending on the fluctuating gas and magnetic flux distributions near the event horizon. We find that our jets reach extremely high efficiencies of ∼100%–300%, even in the absence of an accretion disk. We also calculated the drag forces exerted by the gas onto to the BH and found that the presence of magnetic fields causes the drag forces to be much less efficient than in unmagnetized BHL accretion. They can even sometimes become negative, accelerating the BH rather than slowing it down. Our results extend classical BHL accretion to rotating BHs moving through magnetized media, and demonstrate that accretion and drag are significantly altered in this environment.