For convection dominated problems, we discuss a block Gauss-Seidel method which uses thin cross-wind strips as blocks. The relaxation sweeps of the block Gauss-Seidel method are performed along the the downwind direction. This method is efficient for convection dominated problems, discretized by monotone finite element/finite difference schemes, such as the edge-average finite element method. The use of cross-wind thin blocks is essential to this algorithm, and the algorithm is especially effective when used as a solver instead of a smoother. An algorithm of forming ordering the cross-wind strip blocks is introduced, and the convergence analysis of the proposed block Gauss-Seidel method is performed. Some numerical examples are given to illustrate the effectiveness of the proposed method for convection dominated problems.