An adjoint approach, derived from the reciprocal theorem and the theory of convolution, is used to formulate design sensitivities for linear elastodynamic systems. Variations of a general design functional are expressed in explicit form with respect to variations of all design variables; i.e., material properties, applied loads, prescribed boundary conditions, initial conditions, and shape. The design functional is a function of these explicit variables and the implicit response fields: displacement, velocity, acceleration, stress, strain, and reaction forces. The methodology incorporates the reciprocal theorem between a variation of the real system and its adjoint system. The convolution is employed in lieu of time mappings used in other approaches. In an example problem, the finite element method is utilized to calculate the shape sensitivities of a dynamically loaded plane-strain cantilever plate. These explicit sensitivities are compared with finite difference approximations to validate the formulation and computations.