A flush valve has a relief valve including a stationary relief valve housing and an axially-translatable valve element. The axially-translatable valve element has an axial end that can be selectively translated to unseal an upper opening in a relief passageway through the stationary relief valve housing in order to permit fluid flow therethrough in order to initiate a flush cycle by the lifting of a diaphragm assembly. The axially-translatable valve element may include lead screw-like features which can be used to axially drive the axially-translatable valve element relative to the stationary relief valve housing to open or close the relief valve.