Mixture autoregressive (MAR) models provide a flexible way to model time series with predictive distributions which depend on the recent history of the process and are able to accommodate asymmetry and multimodality. Bayesian inference for such models offers the additional advantage of incorporating the uncertainty in the estimated models into the predictions. We introduce a new way of sampling from the posterior distribution of the parameters of MAR models which allows for covering the complete parameter space of the models, unlike previous approaches. We also propose a relabelling algorithm to deal a posteriori with label switching. We apply our new method to simulated and real datasets, discuss the accuracy and performance of our new method, as well as its advantages over previous studies. The idea of density forecasting using MCMC output is also introduced.