Efficient and robust multigrid solvers for anisotropic problems typically use either semi-coarsened grids or implicit smoothers: line relaxation in 2D and plane relaxation in 3D. However, both of these may be difficult to implement in codes using multi-block structured grids where there may be no natural definition of a global `line' or `plane'. These multi-block structured grids are often used in fluid dynamic applications to capture complex geometries and/or to facilitate parallel processing. In this paper, we investigate the performance of multigrid algorithms using implicit smoothers within the blocks of a such a grid. By looking at a model problem, the 2-D anisotropic diffusion equation, we show that true multigrid efficiency is achieved only when the block sizes are proportional to the strength of the anisotropy. Further, the blocks must overlap and the size of the overlap must again be proportional to the strength of the anisotropy.