The local one-dimensional multisymplectic scheme (LOD-MS) is developed for the three-dimensional (3D) Gross-Pitaevskii (GP) equation in Bose-Einstein condensates. The idea is originated from the advantages of multisymplectic integrators and from the cheap computational cost of the local one-dimensional (LOD) method. The 3D GP equation is split into three linear LOD Schrödinger equations and an exactly solvable nonlinear Hamiltonian ODE. The three linear LOD Schrödinger equations are multisymplectic which can be approximated by multisymplectic integrator (MI). The conservative properties of the proposed scheme are investigated. It is mass-preserving. Surprisingly, the scheme preserves the discrete local energy conservation laws and global energy conservation law if the wave function is variable separable. This is impossible for conventional MIs in nonlinear Hamiltonian context. The numerical results show that the LOD-MS can simulate the original problems very well. They are consistent with the numerical analysis.