This paper is concerned with the problem of predicting the intensity process of an observed doubly stochastic multichannel Poisson process. Under the only hypothesis that the covariance function of the intensity process is separable, recursive algorithms for the computation of the optimal linear filter and predictor are designed. Approximate solutions to the nonlinear filtering and prediction problems are also given. The main advantage of the proposed solutions is that they can be applied to those situations where the intensity process does not satisfy a stochastic differential equation.