Recent studies have demonstrated that periodic time-averaged acoustic fields can be produced from traveling surface acoustic waves (SAWs) in microfluidic devices. This is caused by diffractive effects arising from a spatially limited transducer. This permits the generation of acoustic patterns evocative of those produced from standing waves, but instead with the application of a traveling wave. While acoustic pressure fields in such systems have been investigated, acoustic streaming from diffractive fields has not. In this work we examine this phenomenon and demonstrate the appearance of geometry-dependent acoustic vortices, and demonstrate that periodic, identically rotating Rayleigh streaming vortices result from the imposition of a traveling SAW. This is also characterized by a channel-spanning flow that bridges between adjacent vortices along the channel top and bottom. We find that the channel dimensions determine the types of streaming that develops; while Eckart streaming has been previously presumed to be a distinguishing feature of traveling-wave actuation, we show that Rayleigh streaming vortices also results. This has implications for microfluidic actuation, where traveling acoustic waves have applications in microscale mixing, separation, and patterning.