Sandwich panels, comprising face sheets enclosing a core, are increasingly common structural elements in a variety of applications, including aircraft fuselages and flight surfaces, vehicle panels, lightweight enclosures, and bulkheads. The design flexibility associated with such composite structures provides significant opportunities for tailoring the structure to the load and dynamic response requirements for the particular application. Design flexibility encompasses the details of the face sheets and the core. This paper deals with the numerical optimization of different sandwich configurations for the purposes of achieving reduced structural acoustic response. Laminated face sheets and core geometries, comprising honeycomb and truss-like structures, are considered. The relative importance of the mass and stiffening properties of the core and face sheets are discussed. The optimization work is carried out using commercial codes. Benefits and limits of using an optimization algorithm based on gradient methods are highlighted.