The authors consider the problem of compressive sensed video recovery via iterative thresholding algorithm. Traditionally, it is assumed that some fixed sparsifying transform is applied at each iteration of the algorithm. In order to improve the recovery performance, at each iteration the thresholding could be applied for different transforms in order to obtain several estimates for each pixel. Then the resulting pixel value is computed based on obtained estimates using simple averaging. However, calculation of the estimates leads to significant increase in reconstruction complexity. Therefore, the authors propose a heuristic approach, where at each iteration only one transform is randomly selected from some set of transforms. First, they present simple examples, when block-based 2D discrete cosine transform is used as the sparsifying transform, and show that the random selection of the block size at each iteration significantly outperforms the case when fixed block size is used. Second, building on these simple examples, they apply the proposed approach when video block-matching and 3D filtering (VBM3D) is used for the thresholding and show that the random transform selection within VBM3D allows to improve the recovery performance as compared with the recovery based on VBM3D with fixed transform.