The $S_N$ method has computational simplicity and has been widely used to solve the transport equation. However, it is well known that the ray effect occurs in streaming dominant problems. But the characteristic method gives much more accurate solution than $S_N$ method in the deep penetration problems, since the exiting angular flux is calculated by analytically integrating the transport equation along the characteristic line. In this study, Filippone``s streaming rays(SR) method, a class of characteristic methods is applied to several streaming dominant problems in which the ray effect occurs seriously, with extensions to multigroup, linearly anisotropic scattering, and three-dimensional angular space(x-y-z(infinite)) treatments for the purpose of improving generality and accuracy. For verification, the results of MORSE-CG, obtained with sufficient neutron histories, are used as reference solution. The solution of $S_N$ method is obtained by using the TWODANT code. The results show that the solutions of SR method are in good agreement with those of the MORSE-CG code for most benchmark problems considered, and the ray effect is significantly reduced. In particular, it is noted that the solution of SR method is very accurate in the vacuum duct region. Also, to reduce the computing time, two acceleration methods are applied to the SR method. One is the standard coarse-mesh rebalance acceleration and the other is a new angular two-grid acceleration. The implementation of the angular two-grid acceleration is very simple and the computing time is comparable to that of the coarse-mesh rebalance acceleration. In particular, it is worthwhile to note that the spectral radius of the angular two-grid acceleration is smaller than that of $P_1$ synthetic acceleration for streaming dominant problems($c<0.5$).