Graph-based methods for signal processing have shown promise for the analysis of data exhibiting irregular structure, such as those found in social, transportation, and sensor networks. Yet, though these systems are often dynamic, state-of-the-art methods for graph signal processing ignore the time dimension. To address this shortcoming, this paper considers the statistical analysis of time-varying graph signals. We introduce a novel definition of joint (time-vertex) stationarity, which generalizes the classical definition of time stationarity and the recent definition appropriate for graphs. This gives rise to a scalable Wiener optimization framework for denoising, semi-supervised learning, or more generally inverting a linear operator, that is provably optimal. Experimental results on real weather data demonstrate that taking into account graph and time dimensions jointly can yield significant accuracy improvements in the reconstruction effort.