This paper reports the shock response of layered composites subjected to flyer-plate impact. The composites comprised of Oxygen-Free Copper (Cu) and polymethyl methacrylate layers angled between 0° and 90°. Multi-layered samples were bonded at various orientations, with 0° indicating target layers aligned parallel to the impact direction. At lower angles of orientation, a twofold wave structure consisting of a low-amplitude elastic precursor and a high-amplitude stress wave was observed. The elastic precursor was characterized and influenced by the longitudinal sound speed of Cu and diminished with an increase in sample orientation. As the orientation of the sample increased, an increase in the rise time and a decrease in the wave velocity were recorded. Numerical simulations highlighted the role of the impedance mismatch, as well as geometric dispersion, and oblique interference scattering in layered composites with varying orientations. It is shown that these three factors play a crucial role in shock wave dissipation and dispersion.