Unsupervised speech enhancement based on variational autoencoders has shown promising performance compared with the commonly used supervised methods. This approach involves the use of a pre-trained deep speech prior along with a parametric noise model, where the noise parameters are learned from the noisy speech signal with an expectationmaximization (EM)-based method. The E-step involves an intractable latent posterior distribution. Existing algorithms to solve this step are either based on computationally heavy Monte Carlo Markov Chain sampling methods and variational inference, or inefficient optimization-based methods. In this paper, we propose a new approach based on Langevin dynamics that generates multiple sequences of samples and comes with a total variation-based regularization to incorporate temporal correlations of latent vectors. Our experiments demonstrate that the developed framework makes an effective compromise between computational efficiency and enhancement quality, and outperforms existing methods.