We prove the existence of strong time-periodic solutions and their asymptotic stability with the total energy of the perturbations decaying to zero at an exponential decay rate as $t\to \mathrm{\infty }$ for a semilinear (nonlinearly coupled) magnetoelastic system in bounded, simply connected three-dimensional domain. The mathematical model includes a mechanical dissipation and a periodic forcing function of period $T$. In the second part of the paper, we consider a magnetoelastic system in the form of a semilinear initial-boundary-value problem in a bounded, simply connected two-dimensional domain. We use the LaSalle invariance principle to obtain results on the asymptotic behavior of solutions. This second result was obtained for the system under the action of only one dissipation (the natural dissipation of the system).