In this paper we examine the influence of periodic islands within a time periodic chaotic flow on the evolution of a scalar tracer. The passive scalar tracer is injected into the flow field by means of a steady source term. We examine the distribution of the tracer once a periodic state is reached, in which the rate of injected scalar balances advection and diffusion with the molecular diffusion κ. We study the two-dimensional velocity field u(x,y,t)=2 cos2(ωt)(0,sin x)+2 sin2(ωt)(sin y,0). As ω is reduced from an O(1) value the flow alternates through a sequence of states which are either globally chaotic, or contain islands embedded in a chaotic sea. The evolution of the scalar is examined numerically using a semi-Lagrangian advection scheme. By time-averaging diagnostics measured from the scalar field we find that the time-averaged lengths of the scalar contours in the chaotic region grow like κ−1/2 for small κ, for all values of ω, while the behavior of the time-averaged maximum scalar value, Cmax¯, for small κ depends strongly on ω. In the presence of islands Cmax¯∼κ−α for some α between 0 and 1 and with κ small, and we demonstrate that there is a correlation between α and the area of the periodic islands, at least for large ω. The limit of small ω is studied by considering a flow field that switches from u=(0,2 sin x) to u=(2 sin y,0) at periodic intervals. The small κ limit for this flow is examined using the method of matched asymptotic expansions. Finally the role of islands in the flow is investigated by considering the time-averaged effective diffusion of the scalar field. This diagnostic can distinguish between regions where the scalar is well mixed and regions where the scalar builds up.