The isostructural hydrogensulfides of the alkali metals sodium, potassium, and rubidium were investigated by quasielastic neutron scattering in order to gain insight into the dynamics of the molecular anion HS- as a function of temperature. A phase transition between the rhombohedral middle temperature modification (MTM) and the cubic high temperature modification (HTM) of the title compounds was of special interest. A two site jump model is clearly supported by the data of the MTM and an eight site jump model is consistent with the data of the HTM. The jump distances are in reasonable agreement with distances of the hydrogen positions determined by neutron powder diffraction on the fully deuterated samples. At the MTM --> HTM phase transition the correlation times of the reorientational motion decrease by about one order of magnitude. Within the investigated temperature range (T=170 K to T=450 K) they change from 10(-8) s to 10(-12) s. They show Arrhenius behavior. The activation enthalpies of the MTM decrease with increasing radius of the cation from 26(1) kJ mol(-1), 22(1) kJ mol(-1) to 18(2) kJ mol(-1) for NaHS, KHS and RbHS respectively. In the HTM they are equal within the estimated error 6(1) kJ mol(-1) and 7(1) kJ mol(-1) for NaHS and KHS.