Could electricity work with a flow of particles other than electrons? Would a flow of positrons be theoretically possible? A flow of protons?
 A: Proton conduction happens in several types of oxides, ceramics and crystals, due to the fact that the hydrogen nucleus is small enough to move around the lattices with relatively small resistance. In some cases the material might behave as an insulator for electron currents but as a conductor for proton, in which case proton diffusion becomes the dominant factor in electric conductance
A: A flow of positrons is possible in a conductor made of anti-matter and would definitely constitute electricity. However, a flow of positrons in a normal matter is not possible, as they would instantly annihilate with electrons. And because there is no antimatter anywhere around, this remains only a theoretical possibility at the moment. A flow of positrons is possible in a vacuum though.
A flow of protons is not practical to produce electricity in a solid conductor, but is possible in a vacuum or a conductor made of an ionized gas (such as hydrogen $H_2\rightarrow 2p^++2e^-$) or during an electrolysis in a liquid. Solar flares often send a stream of protons (as well as electrons) our way. Their current is deflected by the Earth magnetic field to the poles where they produce spectacular auroras.
During the electrolysis, water splits into a proton and a negative ion both conducting electricity: $H_2O\rightarrow p^++OH^-$. So proton currents are not only possible, but are very common.
