Estimation of first excursion probabilities is one of the most challenging problems in stochastic structural dynamics. Hence, this contribution presents a framework for evaluating first excursion probabilities for a particular class of problems, i.e. linear systems whose structural parameters are characterized as uncertain and which are subjected to discrete white noise excitation. The focus of the proposed approach is on problems involving a large number of uncertain parameters associated with both excitation and structural parameters. The sought probability is estimated using Importance Sampling (IS). The Importance Sampling density (ISD) function associated with the uncertain parameters exploits the linearity of the problem at hand. For generating samples distributed according to the ISD function, the so-called Sampling-Importance Resampling (SIR) is applied in combination with a surrogate model for the spectral properties of the structure. This surrogate model is applied in order to improve numerical efficiency. An application example comprising a FE model illustrates the advantages of the proposed approach.