A discrete-time approximation scheme called local linearization of the Langevin diffusion on $R^k$ is considered, with emphasis on the ergodic properties of the approximation considered as a discrete-time Markov chain. We will derive criteria for the scheme to be geometrically ergodic, and illustrate the use of these criteria by means of examples. Furthermore, we discuss the scheme in relation to other schemes and the use of such discretization schemes as proposals in a Metropolis-Hastings algorithm.