We examine the transport behaviour of non-interacting particles in a simple channel billiard, at equilibrium and in the presence of an external field. The channel walls are constructed from straight line-segments. We observe a sensitive dependence on the model parameters of the transport properties, which range from sub-diffusive to super-diffusive regimes. In non-equilibrium, we find a transition in the transport behaviour between seemingly-chaotic and (quasi-) periodic behaviour. Our results support the view that normal transport laws do not need chaos, or quenched disorder, to be realized. Furthermore, they motivate some new definitions of complexity, which are relevant for transport phenomena.