The paths followed by individual fluid particles can be extremely complicated even in smooth laminar flows. Such chaotic advection causes mixing of the fluid. This phenomenon is studied analytically for a class of spatially periodic flows comprising a basic flow of two-dimensional (or axisymmetric) counter-rotating vortices in a layer of fluid, and modulated by a perturbation which is periodic in time and/or space. Examples of this type of flow include Bénard convection just above the point of instability of two-dimensional roll cells, and Taylor vortex flow between concentric rotating cylinders. The transport of chaotically advected particles is modelled as a Markov process. This predicts diffusion-like mixing, and provides an expression for the diffusion coefficient. This expression explains some features of experimental results reported by Solomon & Gollub (1988): its accuracy is investigated through a detailed comparison with numerical results from a model of wavy Taylor vortex flow. The approximations used in the analysis are equivalent to those used to obtain the quasi-linear result for diffusion in the standard map.