<p>Randomized methods can be competitive for the solution of problems with a large matrix of low rank. This paper explores their application to the solution of linear discrete ill-posed problems whose matrices often can be approximated well by a matrix of fairly low rank. Recently, a method based on randomized SVD and Tikhonov regularization was found to be competitive with a Krylov method based on a partial Golub-Kahan bidiagonalization with regard to both speed and quality of the computed solution when applied to linear discrete ill-posed problems in one space-dimension. However, for problems in two space-dimensions, the randomized methods often failed to deliver solutions of satisfactory quality unless the singular values of the matrix decreased to numerical zero very quickly with increasing index. This work proposes the application of randomized block Krylov methods to the solution of linear discrete ill-posed problems in one and two space-dimensions. Computed examples illustrate their competitiveness both in terms of the quality of the computed solution and computing time.</p>