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Integral field spectroscopy of high-redshift galaxies has become a powerful tool for understanding their dynamics and evolutionary states. However, in the case of gravitationally lensed systems, it has proved difficult to model both lensing and intrinsic kinematics in a way that takes full advantage of the information available in the spectral domain. In this paper, we introduce a new method for pixel-based source reconstruction that alters standard regularization schemes for two-dimensional (2D) data in a way that leverages kinematic information in a physically motivated but flexible fashion, and that is better suited to the three-dimensional (3D) nature of integral field data. To evaluate the performance of this method, we compare its results to those of a more traditional 2D nonparametric approach using mock Atacama Large Millimeter/submillimeter Array (ALMA) observations of a typical high-redshift dusty star-forming galaxy. We find that 3D regularization applied to an entire data cube reconstructs a source’s intensity and velocity structure more accurately than 2D regularization applied to separate velocity channels. Cubes reconstructed with 3D regularization also have more uniform noise and resolution properties and are less sensitive to the signal-to-noise ratio of individual velocity channels than the results of 2D regularization. Our new approach to modeling integral field observations of lensed systems can be implemented without making restrictive a priori assumptions about intrinsic kinematics, and opens the door to new observing strategies that prioritize spectral resolution over spatial resolution (e.g., for multiconfiguration arrays like ALMA).