A viscoelastic model is established for the coupled motion in a thin solid rod and an infinite viscous medium surrounding it, starting from the basic laws of elastic and fluid mechanics. The effect of various material parameters of the surrounding medium on the propagation of acoustic waves along the rod is investigated using numerical calculations. It is demonstrated that, at low frequencies, the fundamental radial–axial mode (R01) responds most to changes in the material parameters of the surrounding medium related to shear motion in a manner similar to that of the fundamental torsional mode (T00). Based on this effect, a model miniature viscosity and density sensor is proposed and tested in viscous liquids. The attenuation in the device agrees well with the theoretical model and exhibits an approximately linear dependence on the square root of the product of the density and the shear viscosity of the surrounding medium. This work shows that simple traditional acoustic technology can be used to develop small size, low‐cost sensors for liquid viscosity and density measurement with the possibility of on‐line applications.