We explore the flow of a nematic liquid crystal in microfluidic channels with a rectangular cross section through experiments and numerical modeling. The flow profile and the liquid crystal orientational profile show three distinct regimes of weak, medium, and strong flow as the driving pressure is varied. These are identified by comparing polarizing optical microscopy experiments and numerical solutions of the nematofluidic equations of motion. The relative stability of the regimes is related to the de Gennes characteristic shear-flow lengths e(1) and e(2), together with the channel's aspect ratio w/d. Finally, we show that the liquid crystalline microfluidic flow can be fully steered from left to right of a simple microchannel by applying transverse temperature gradients.