The heavier alkali metal hydrides MH (M = K, Rb, Cs) undergo a series of pressure induced structural phase transitions: B1 (NaCl) → B2 (CsCl) → CrB. Experiments reveal that the latter occurs at 85 and 17.5 GPa for RbH and CsH, but it has not yet been observed for KH. Herein, evolutionary algorithms coupled with density functional theory calculations are employed to explore the potential energy surface of the aforementioned hydrides up to pressures of 300 GPa. The computations support previous theoretical work which predicts that KH will adopt the CrB structure when compressed. In addition, for KH and RbH we find configurations with Pnma and I41/amd symmetry that are thermodynamically competitive with the CrB structure at 300 GPa. Between 100–150 GPa, a Pnma structure which is analogous to a high-pressure form of CsI is found to be the most stable phase for the heaviest alkali hydride considered. At higher pressures a hitherto unknown CsH–P63/mmc arrangement becomes thermodynamically preferred up to at least 400 GPa. A detailed analysis of the geometric and electronic structures of the various phases is provided.