A heterogeneous transport solver in three-dimensional (3-D) geometry, GENESIS, is developed incorporating recent developments in the method of characteristics (MOC) in 3-D geometry. The Legendre Polynomial Expansion of Angular Flux (LEAF) method is implemented in the GENESIS code, in which neutron transport is calculated in two-dimensional (2-D) characteristics planes rather than in one-dimensional characteristics lines adopted in the conventional approach of 3-D MOC. Unlike the planar MOC method that combines 2-D MOC calculations through axial leakages, the GENESIS code explicitly considers angular and spatial dependence of outgoing and incoming angular fluxes between axial planes. Thus, the GENESIS code eliminates a crucial approximation used in the planar MOC method: No approximation is used for axial leakage treatment. The GENESIS code can handle flexible shapes of objects in rectangular or hexagonal geometry. A two-level, multigroup generalized coarse mesh rebalance acceleration method is adopted for efficient convergence of neutron transport calculation. Performance of the GENESIS code is verified through various benchmark calculations. The calculation results indicate the fidelity of the GENESIS code based on the LEAF method.