A novel method, the discrete cones method (DCN), is proposed as an alternative to the discrete ordinates method (SN) for solutions of the two-dimensional neutron transport equation. The new method utilizes a new concept, discrete cones, which are made by partitioning a unit spherical surface that the direction vector of particles covers. In this method particles in a cone are simultaneously traced instead of those in discrete directions so that an anomaly of the SN method, the ray effects, can be eliminated. The DCN method has been formulated for X-Y geometry and a program has been created by modifying the standard SN program TWOTRAN-II. Our sample calculations demonstrate a strong mitigation of the ray effects without a computing cost penalty.