Model-X knockoffs is a wrapper essentially transforming any importance measure into a variable selection algorithm, which discovers true effects while rigorously controlling the expected fraction of false positives. A frequently discussed challenge to apply this method is to construct knockoff variables, which are synthetic variables obeying a crucial exchangeability property with the explanatory variables under study. This paper introduces techniques for knockoff generation in great generality: we provide a sequential characterization of all possible knockoff distributions, which leads to an MCMC formulation of an exact knockoff sampler. We further show how to use conditional independence structure to speed up computations. Combining these two threads, we introduce an explicit set of powerful MCMC algorithms including the multiple-try Metropolis and empirically demonstrate their effectiveness. The techniques we develop are sufficiently rich to enable knockoff sampling in challenging models including cases where the covariates are continuous and heavy tailed, and where they follow an Ising model.