In this work, new method is introduced for the adaptive processing of 3D images and multidimensional signals, based on their tensor representations. This is offered a kind of local averaging technique, which uses a recursive sliding mean three-dimensional filter (3DSMF). The investigation is additionally focused on the evaluation of the method effectiveness, based on its computational complexity (CC). The filter performance is presented in detail for the case of adaptive filtration of Gaussian noise. Special attention in the investigation is paid to the specific qualities of the filter working window. For the evaluation of the computational complexity is analyzed the 3D filter implementation through three recursive sliding 1D filters, that run sequentially one after the other. The computational complexity of the recursive version of the filter is compared with that of the non-recursive, and the advantages of the first approach are proved. For the evaluation, a number of basic mathematical operations are needed for the filter performance. The proposed recursive 3DSMF suits various contemporary applications, which comprise 3D convolutional neural networks (CNN) with sliding locally adaptive 3D filtration in their layers. The low CC of the 3DSMF opens wide opportunities for efficient processing, analysis, and visualization of 3D images and multidimensional information through tensor neural networks.