Rainfall-induced landslides have a complex mechanism involving the interaction between the ground structure and rainwater, which makes predicting such events particularly challenging. Although several numerical methods have been developed to predict landslide occurrence and the extent of sediment flow, analysing the entire landslide process in three dimensions in a unified manner remains challenging due to the limited computational efficiency and difficulties in representing deformation and flow. In this study, we present a numerical method for simulating rainfall-induced landslides, which employs the coupled hydromechanical material point method (MPM) for unsaturated porous media based on implicit and explicit formulations. The proposed method enables the efficient analysis of all rainfall-induced landslide processes, including both quasi-static and dynamic processes. The proposed method includes an implicit MPM based on a simplified formulation for unsaturated porous media, which is first applied to the quasi-static analysis in the pre-failure stages where rainwater infiltrates the ground. The explicit coupled hydromechanical MPM is then applied to the dynamic analysis for post-failure stages where the ground collapses and flows. A constitutive law for soils is improved in the simulation of landslide initiation and sediment flow by incorporating the effect of cohesion in a visco-plastic model for granular materials. The performance of the proposed method is demonstrated through simulations using a three-dimensional terrain model based on a digital elevation model (DEM) of Ashikita town, Kumamoto, Japan, where an actual landslide occurred owing to heavy rainfall on July 3-4, 2020. The numerical results obtained from the proposed coupled hydromechanical MPM are compared with those obtained from the single-phase MPM using a strength reduction method. These results indicate that the presence of pore water plays a crucial role in comprehending all rainfall-induced landslide processes, i.e., from the initiation of the landslide to the discharge of sediment.