Monte Carlo (MC) is widely used for the simulation of discrete time Markov chains. We consider the case of a $d$-dimensional continuous state space and we restrict ourselves to chains where the $d$ components are advanced independently from each other, with $d$ random numbers used at each step. We simulate $N$ copies of the chain in parallel, and we replace pseudorandom numbers on $I^d := (0,1)^d$ with stratified random points over $I^{2d}$: for each point, the first $d$ components are used to select a state and the last $d$ components are used to advance the chain by one step. We use a simple stratification technique: let $p$ be an integer, then for $N=p^{2d}$ samples, the unit hypercube is dissected into $N$ hypercubes of measure $1/N$ and there is one sample in each of them. The strategy outperforms classical MC if a well-chosen multivariate sort of the states is employed to order the chains at each step. We prove that the variance of the stratified sampling estimator is bounded by $\mathcal{O}(N^{-(1+1/(2d))})$, while it is $\mathcal{O}(N^{-1})$ for MC. In numerical experiments, we observe empirical rates that satisfy the bounds. We also compare with the Array-RQMC method.