A methodology for the computation of unsteady Euler flows in configurations with moving boundaries is presented. The procedure is composed of a moving and adaptive grid management algorithm and a Lagrangian-Eulerian flow solver. The domain discretization is handled via unstructured triangular grids. The flow scheme uses a generalized version of Roe's approximate Riemann solver, which takes into account the mesh movement while satisfying in an intrinsic way the geometric conservation laws. Grid-flow coupling is obtained through error estimation. The methodology is applied to various flow problems showing moving shock waves and moving geometries.