While the problem of random loads moving across a beam has been studied extensively, the existing methods are limited in terms of their applicability. The most common approach uses modal superposition, which requires that the load be represented in terms of the structure's mode shapes, making the representation of the load intrinsically a function of the structure to which it is applied. To address this problem, this paper approximates the discrete stochastic moving load as a white noise passed through filters constructed with Padé approximants. The resulting model of the load is independent of the structure and enables efficient random vibration methods to be applied for solution of the problem. By representing the structure and the loading together in an augmented state space system, the variance of responses can be found directly and accurately, enabling the traffic-induced responses of a broader class of bridge structures to be analyzed. The effectiveness and usefulness of the proposed approach is demonstrated through two examples of bridge structures.