A fully Bayesian treatment of complicated predictive models (such as deep neural networks) would enable rigorous uncertainty quantification and the automation of higher-level tasks including model selection. However, the intractability of sampling Bayesian posteriors over many parameters inhibits the use of Bayesian methods where they are most needed. Thermodynamic computing has emerged as a paradigm for accelerating operations used in machine learning, such as matrix inversion, and is based on the mapping of Langevin equations to the dynamics of noisy physical systems. Hence, it is natural to consider the implementation of Langevin sampling algorithms on thermodynamic devices. In this work we propose electronic analog devices that sample from Bayesian posteriors by realizing Langevin dynamics physically. Circuit designs are given for sampling the posterior of a Gaussian-Gaussian model and for Bayesian logistic regression, and are validated by simulations. It is shown, under reasonable assumptions, that the time-complexity of sampling the Gaussian-Gaussian posterior is sublinear in dimension. These results highlight the potential to accelerate Bayesian inference with thermodynamic computing.