Jet breakup is an important behavior at a core disruptive accident for a sodium-cooled fast reactor. The lattice Boltzmann (LB) method is adopted to simulate the jet breakup behavior. The Multiple-Relaxation Time (MRT) scheme is introduced into the existing three-dimensional 19-velocity (D3Q19) LB model for immiscible two-phase flow to enhance the numerical stability for low kinematic viscosity. The simulation results show that the present LB model using MRT enables to simulate the jet breakup behavior, where the kinematic viscosity is of the order of 10^-3. The velocity field and interfacial shape are compared with the experimental result using PIV and Laser-Induced Fluorescence (LIF). The interfacial instability and fragmentation behavior of the jet can be also simulated. Comparison of the LB simulation with experimental data shows that the time series of jet leading edge can be simulated within an error of around 10%.