This paper describes and compares some structure preserving techniques for the solution of linear discrete ill-posed problems with the t-product. A new randomized tensor singular value decomposition (R-tSVD) with a t-product is presented for low tubal rank tensor approximations. Regularization of linear inverse problems by truncated tensor eigenvalue decomposition (T-tEVD), truncated tSVD (T-tSVD), randomized T-tSVD (RT-tSVD), t-product Golub-Kahan bidiagonalization (tGKB) process, and t-product Lanczos (t-Lanczos) process are considered. A solution method that is based on reusing tensor Krylov subspaces generated by the tGKB process is described. The regularization parameter is the number of iterations required by each method. The discrepancy principle is used to determine this parameter. Solution methods that are based on truncated iterations are compared with solution methods that combine Tikhonov regularization with the tGKB and t-Lanczos processes. Computed examples illustrate the performance of these methods when applied to image and gray-scale video restorations. Our new RT-tSVD method is seen to require less CPU time and yields restorations of higher quality than the T-tSVD method.