Magnetically actuated microswimmers have recently attracted attention due to many possible biomedical applications. In this study we investigate the dynamics of rigid magnetically rotated microswimmers with permanent magnetic dipoles. Our approach uses a boundary element method to calculate a mobility matrix, accurate for arbitrary geometries, which is then used to identify the steady periodically rotating orbits in a co-rotating body-fixed frame. We evaluate the stability of each of these orbits. We map the magnetoviscous behavior as a function of dimensionless Mason number and as a function of the angle that the magnetic field makes with its rotation axis. We describe the wobbling motion of these swimmers by investigating how the rotation axis changes as a function of experimental parameters. We show that for a given magnetic field strength and rotation frequency, swimmers can have more than one stable periodic orbit with different rotation axes. Finally, we demonstrate that one can improve the controllability of these types of microswimmers by adjusting the relative angle between the magnetic field and its axis of rotation.