We investigate the behavior of soft magnetoactive periodic laminates under remotely applied magnetic field. We derive explicit formulae for the induced deformation due to magnetic excitation of the laminates with hyperelastic magnetoactive phases. Next, we obtain the closed-form formulae for the velocities of long transverse waves. We show the dependence of the wave velocity on the applied magnetic intensity and induced strains, as well as on the wave propagation direction. Based on the long wave analysis, we derive closed-form formulae for the critical magnetic field corresponding to the loss of macroscopic stability. Finally, we analyze the transverse wave band gaps appearing in magnetoactive laminates in the direction normal to the layers. We illustrate the band gap tunability—width and position—by magnetically induced deformation.