A multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is proposed for simulating flow and heat transfer in the hydrodynamically and thermally anisotropic porous medium at the representative-elemen tary-volume (REV) scale. By selecting the appropriate equilibrium distributions, relaxation matrix and discrete force/heat source terms, the present MRT LB model can recover the correct Darcy-Brinkman-Forchheimer and energy equations with anisotropic permeability and thermal conductivity through the Chapman-Enskog procedure. Several natural convection problems in anisotropic porous medium are simulated to validate the present LB model. The corresponding numerical results are in good agreement with data in the available literature. Especially, natural convection in a cavity with two anisotropic porous layers is investigated. The numerical results indicates that, the use of anisotropic porous layer with some optimal parameters can produce higher rate of heat transfer compared with the isotropic porous layer. (C) 2016 Elsevier Ltd. All rights reserved.