We show that a surface acoustic wave (SAW) applied across the terminals of a magnetic tunnel junction (MTJ) decreases both the (time-averaged) parallel and antiparallel resistances of the MTJ, with the latter decreasing much more than the former. This results in a decrease of the tunneling magnetoresistance (TMR) ratio. The coercivities of the free and fixed layer of the MTJ, however, are not affected significantly, suggesting that the SAW does not cause large-angle magnetization rotation in the magnetic layers through the inverse magnetostriction (Villari) effect at the power levels used. This study sheds light on the dynamical behavior of an MTJ under periodic compressive and tensile strain.