This version has replaced the version of January 30, 2012.
A successful construction of an importance density for nonlinear non-Gaussian state space models is crucial when Monte Carlo simulation methods are used for likelihood evaluation, signal extraction of dynamic latent factors and forecasting. The method of efficient importance sampling is successful in this respect but we show that it can be implemented more conveniently using standard Kalman filter and smoothing methods. We further obtain computational gains by simulating directly from the signal equation rather than simulating from the usually higher dimensional state equation. Our results provide some new insights but they primarily lead to a more simple and fast method for efficient importance sampling. In a simulation study we provide some evidence of the computational gains. Our new methods are illustrated for a stochastic volatility model with a Student's t distribution.