We present a numerical technique for computing flowfields around moving solid boundaries immersed in flows on fixed meshes. The mixed Eulerian-Lagrangian framework treats the immersed boundaries as sharp interfaces and a finite volume formulation for the flow solver allows boundary conditions at the moving surfaces to be exactly applied. A second-order accurate spatial and temporal discretization is employed with a fractional-step scheme for solving the flow equations. A multigrid accelerator for the pressure Poisson equations has been developed to apply in the presence of multiple embedded solid regions on the mesh. We validate the numerics by comparing against experimental and numerical results on two problems: 1) The flow in a channel with a moving indentation in one wall and 2) The dynamics of vortex shedding from a cylinder oscillating in the free-stream.