The propagation of viscous, thin gravity currents of non-Newtonian liquids in horizontal and inclined channels with semicircular and triangular cross-sections is investigated theoretically and experimentally. The liquid rheology is described by a power-law model with flow behaviour index $n$, and the volume released in the channel is taken to be proportional to $t^{α}$, where $t$ is time and $α$ is a non-negative constant. Some results are generalised to power-law cross-sections. These conditions are representative of environmental flows, such as lava or mud discharges, in a variety of conditions. Theoretical solutions are obtained in self-similar form for horizontal channels, and with the method of characteristics for inclined channels. The position of the current front is found to be a function of the current volume, the liquid rheology, and the channel inclination and geometry. The triangular cross-section is associated with the fastest or slowest propagation rate depending on whether $α<α_c$ or $α>α_c$. For horizontal channels, $α_c=n/(n+1)<1$, whereas for inclined channels, $α_c=1$, irrespective of the value of $n$. Experiments were conducted with Newtonian and power-law liquids by independently measuring the rheological parameters and releasing currents with constant volume ($α=0$) or constant volume flux ($α=1$) in right triangular and semicircular channels. The experimental results validate the model for horizontal channels and inclined channels with $α=0$. For tests in inclined channels with $α=1$, the propagation rate of the current front tended to lower values than predicted, and different flow regimes were observed, i.e., uniform flow with normal depth or instabilities resembling roll waves at an early stage of development. The theoretical solution accurately describes current propagation with time before the transition to longer roll waves. An uncertainty analysis reveals that the rheological parameters are the main source of uncertainty in the experiments and that the model is most sensitive to their variation. This behaviour supports the use of carefully designed laboratory experiments as rheometric tests.