The efficient computation of transient fields radiated by non-harmonic source distributions is a problem relevant in numerous areas of acoustics. This paper presents an efficient easily implemented method for the generation of time-dependent spherical harmonic expansions for arbitrary sources, which can be used to compute the transient radiated field at arbitrary points outside the source domain. The method depends on the theory of time-domain spherical harmonic expansions and the solution of Vandermonde systems. Results are presented demonstrating the efficiency and accuracy of the method with respect to full evaluation of the field radiated by a randomized source distribution.