OptoMechanical Modulation Tomography (OMMT) exploits compressed sensing to reconstruct high resolution microscopy volumes from fewer measurement images compared to exhaustive section sampling in conventional light sheet microscopy. Nevertheless, the volumetric reconstruction process is computationally expensive, making it impractically slow to use on large-size images, and prone to generating visual artefacts. Here, we propose a reconstruction approach that uses a 1+2D Total Variation (TV1+2) regularization that does not generate such artefacts and is amenable to efficient implementation using parallel computing. We evaluate our method for accuracy and scaleability on simulated and experimental data. Using a high quality, but computationally expensive, Plug-and-Play (PnP) method that uses the BM4D denoiser as a benchmark, we observe that our approach offers an advantageous trade-off between speed and accuracy. (c) 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement