Diffraction of a three-dimensional (3D) spatiotemporal optical pulse by a phase-shifted Bragg grating (PSBG) is considered. The pulse diffraction is described in terms of signal transmission through a linear system with a transfer function determined by the reflection or transmission coefficient of the PSBG. Resonant approximations of the reflection and transmission coefficients of the PSBG as functions of the angular frequency and the in-plane component of the wave vector are obtained. Using these approximations, a hyperbolic partial differential equation (Klein-Gordon equation) describing a general class of transformations of the incident 3D pulse envelope is derived. A solution to this equation is found in the form of a convolution integral. The presented rigorous simulation results fully confirm the proposed theoretical description. The obtained results may find application in the design of new devices for spatiotemporal pulse shaping and for optical information processing and analog optical computing.