In this paper, the local linearization method for the approximate computation of the prediction and filtering estimates of continuous-discrete state space models is extended to the general case of non-linear non-autonomous models with multiplicative noise. The approximate prediction and filter estimates are obtained by applying the optimal linear filter to the piecewise linear state space model that emerges from a local linearization of both the non-linear state equation and the non-linear measurement equation. In addition, the solutions of the differential equations that describe the evolution of the first two conditional moments between observations are obtained, and an algorithm for their numerical computation is also given. The performance of the LL filters is illustrated by mean of numerical experiments.