Many micro and nano-technologies rely upon the complicated motion of objects immersed in a viscous fluid. It is often the case that for such problems analytical theory is not available to quantitatively describe and predict the device dynamics. In addition, the numerical simulation of such devices involves moving boundaries and use of the standard Eulerian computational approaches are often difficult to implement. In order to address this problem we use and validate a fully Lagrangian finite element approach that treats the moving boundaries in a natural manner. We validate the method for use in calculating the dynamics of oscillating objects in a viscous fluid. Specifically, the dynamics of a micron-scale cylinder oscillating in water are studied numerically. The fluid dynamics generated by an infinitely long cylinder are a good approximation for the flow field around an oscillating cantilever. The numerical results agree well with analytical theory. It is anticipated that further development of the fully Lagrangian numerical approach for fluid-solid interaction problems will be useful in the development of micro and nano-technologies.