We consider source seeking tasks, where the goal is to locate a source using a mobile agent that gathers potentially noisy measurements of the signal emitted by the source. This arises, for example, when searching for a radioactive or chemical source using mobile physical sensors that detect particles carried by the wind. In this work, we propose an iterative Bayesian algorithm for source seeking, especially well-suited for challenging environments where the signal intensity is multimodal and observations are noisy. At every step, our algorithm computes a Bayesian posterior distribution over the source location using prior physical knowledge of the observation process along with the observations collected so far. Then, it decides where the agent should move and observe next by following a search strategy that implicitly considers paths to the source's most likely location under the posterior. We show that the trajectory of an agent executing the proposed algorithm converges to the source location asymptotically with probability one. We validate the algorithm's convergence through simulated experiments of an agent seeking a source of a chemical plume in a turbulent environment.