We introduce MCCE: Monte Carlo sampling of valid and realistic Counterfactual Explanations for tabular data, a novel counterfactual explanation method that generates on-manifold, actionable and valid counterfactuals by modeling the joint distribution of the mutable features given the immutable features and the decision. Unlike other on-manifold methods that tend to rely on variational autoencoders and have strict prediction model and data requirements, MCCE handles any type of prediction model and categorical features with more than two levels. MCCE first models the joint distribution of the features and the decision with an autoregressive generative model where the conditionals are estimated using decision trees. Then, it samples a large set of observations from this model, and finally, it removes the samples that do not obey certain criteria. We compare MCCE with a range of state-of-the-art on-manifold counterfactual methods using four well-known data sets and show that MCCE outperforms these methods on all common performance metrics and speed. In particular, including the decision in the modeling process improves the efficiency of the method substantially.