This paper describes laminar fluid flow through a threedimensional elastomeric microstructure formed by two microfluidic channels, fabricated in layers that contact one another face-to-face (typically at a 90° angle), with the fluid flows in tangential contact. There are two ways to control fluid flow through these tangentially connected microchannels. First, the flow profiles through the crossings are sensitive to the aspect ratio of the channels; the flow can be controlled by applying external pressure and changing this aspect ratio. Second, the flow direction of an individual laminar stream in multiphase laminar flow depends on the lateral position of the stream within the channel; this position can be controlled by injecting additional streams of fluid into the channel. We describe two microfluidic switches based on these two ways for controlling fluid flow through tangential microchannels and present theoretical arguments that explain the observed dependence of the flow profiles on the aspect ratio of the channels.