The "standard" numerical methods used for inverting the Laplace transform are based on a regularization of an exact inversion formula. They are very sensitive to noise in the Laplace transformed function. In this article we suggest a different strategy. The inversion formula we use is an approximate one, but it is stable with respect to noise. The new approximate expression is obtained from a short time expansion of the Bromwich inversion formula. We show that this approximate result can be significantly improved when iterated, while remaining stable with respect to noise. The iterated method is exact for the class of functions of type E(m)e(aE). The method is applied to a harmonic model of the stilbene molecule, to a truncated exponent series, and to the flux-flux correlation function for the parabolic barrier. These examples demonstrate the utility of the method for application to problems of interest in molecular dynamics.