The two alkali silanides ASiH(3) (A = K and Rb) were investigated by means of quasielastic neutron scattering, both below and above the order-disorder phase transition occurring at around 275-300 K. Measurements upon heating show that there is a large change in the dynamics on going through the phase transition, whereas measurements upon cooling reveal a strong hysteresis due to undercooling of the disordered phase. The results show that the dynamics is associated with rotational diffusion of SiH3- anions, adequately modeled by H-jumps among 24 different jump locations radially distributed around the Si atom. The average relaxation time between successive jumps is of the order of subpicoseconds and exhibits a weak temperature dependence with a small difference in activation energy between the two materials, 39(1) meV for KSiH3 and 33(1) meV for RbSiH3. The pronounced SiH3- dynamics explains the high entropy observed in the disordered phase resulting in the low entropy variation for hydrogen absorption/desorption and hence the origin of these materials' favorable hydrogen storage properties.