We report a novel technique capable of measuring the kinematic shear viscosity of Newtonian liquids with steady streaming in microfluidic devices. This probe-free microrheological method utilizes sub-kilohertz liquid oscillation frequencies around a cylindrical obstacle, ensuring that the inner streaming layer is comparable in size to the cylinder radius. To calibrate the viscometer, the evolution of the inner streaming layer as a function of oscillation frequency for a liquid of known viscosity is characterized using standard particle tracking techniques. Once calibrated, we show how the steady streaming viscometer can be used to measure low-viscosity liquids and volatile liquids.