Nonreciprocity, the defining characteristic of isolators, circulators, and a wealth of other applications in radio/microwave communications technologies, is generally difficult to achieve as most physical systems incorporate symmetries that prevent the effect. In particular, acoustic waves are an important medium for information transport, but they are inherently symmetric in time. In this work, we report giant nonreciprocity in the transmission of surface acoustic waves (SAWs) on lithium niobate substrate coated with ferromagnet/insulator/ferromagnet (FeGaB/Al2O3/FeGaB) multilayer structure. We exploit this structure with a unique asymmetric band diagram and expand on magnetoelastic coupling theory to show how the magnetic bands couple with acoustic waves only in a single direction. We measure 48.4-dB (power ratio of 1:69,200) isolation that outperforms current state-of-the-art microwave isolator devices in a previously unidentified acoustic wave system that facilitates unprecedented size, weight, and power reduction. In addition, these results offer a promising platform to study nonreciprocal SAW devices.