In this paper, we develop an efficient numerical method based on Gauss-Hermite quadrature to calculate the one-dimensional dynamic response of a nonlocal, peridynamic bar composed of (inhomogeneous) linear material. The principal physical characteristic of the peridynamic theory is the presence of long-range forces leading to nonlinear dispersion relations while the principal mathematical characteristic is the presence of a stationary Barbashin operator in the integro-differential equation of motion. We calculate two examples corresponding to continuous and discontinuous, Riemann-like initial conditions. As the analytical solutions for these examples are known they serve as validation problems for the proposed numerical scheme.