The ionic conductivity has been investigated within a wide temperature and concentration range for the rotator phase fcc-Li2SO4 doped with 0 to 8 mol% MgSO4. For comparison we also measured the ionic conductivity in the melt. For the solid phase the temperature dependence of the conductivity deviates from the Arrhenius behaviour, but it can be successfully fitted to the Vogel-Tammann-Fulcher (VTF) equation. Also for the bcc high temperature rotator phases of LiNaSO4 and LiAgSO4 we found that the VTF equation provides a better fit than the Arrhenius equation. The ideal glass transition temperature obtained from the fit to the VTF equation is (166+20)degrees C for the fee phase of Li2SO4-MgSO4, independent of the MgSO4 concentration. For the bcc-phases of LiNaSO4 and LiAgSO4 it is (113+/-19)degrees C and (135+/-49)degrees C respectively. This indicates that although these materials are crystalline, the ionic conductivity behaves in a similar way to that of an amorphous material above the glass transition temperature. For comparison we also reevaluated ionic conductivity data over a wide temperature range for the archetypical crystalline solid electrolyte, alpha-AgI. As expected, the Arrhenius equation provides an excellent fit in this case since the ionic transport can be described as ionic hopping between available sites. For the high temperature rotator phases the dynamical disorder, due to the rotational freedom of the sulphate ions, is thought to be a key factor for the ionic transport. It is thus suggested that the free volume model may be used to describe the ionic transport in high temperature rotator phases.