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Abstract: This paper presents a Bayesian hierarchical spatiotemporal method of interpolation, termed as Markov Cube Kriging (MCK). The classical Kriging methods become computationally prohibitive, especially for large datasets due to the O(n3) matrix decomposition. MCK offers novel and computationally efficient solutions to address spatiotemporal misalignment, mismatch in the spatiotemporal scales and missing values across space and time in large spatiotemporal datasets. MCK is flexible in that it allows for non-separable spatiotemporal structure and nonstationary covariance at the hierarchical spatiotemporal scales. Employing MCK we developed estimates of daily concentration of fine particulates matter ≤2.5 μm in aerodynamic diameter (PM2.5) at 2.5 km spatial grid for the Cleveland Metropolitan Statistical Area, 2000 to 2009. Our validation and cross-validation suggest that MCK achieved robust prediction of spatiotemporal random effects and underlying hierarchical and nonstationary spatiotemporal structure in air pollution data. MCK has important implications for environmental epidemiology and environmental sciences for exposure quantification and collocation of data from different sources, available at different spatiotemporal scales. [Copyright &y& Elsevier]