Due to the limitations of current voltage sensing techniques, optimal filtering of noisy, undersampled voltage signals on dendritic trees is a key problem in computational cellular neuroscience. These limitations lead to voltage data that is incomplete (in the sense of only capturing a small portion of the full spatiotemporal signal) and often highly noisy. In this paper we use a Kalman filtering framework to develop optimal experimental design methods for voltage sampling. Our approach is to use a simple greedy algorithm with lazy evaluation to minimize the expected square error of the estimated spatiotemporal voltage signal. We take advantage of some particular features of the dendritic filtering problem to efficiently calculate the Kalman estimator’s covariance. We test our framework with simulations of real dendritic branching structures and compare the quality of both time-invariant and time-varying sampling schemes. While the benefit of using the experimental design methods was modest in the time-invariant case, improvements of 25-100% over more naive methods were found when the observation locations were allowed to change with time. We also present a heuristic approximation to the greedy algorithm that is an order of magnitude faster while still providing comparable results.