This paper presents an efficient numerical method for linear elasticity problem on complex domains in three dimensions. It solves the linear elasticity problem in the framework of boundary integral equations (BIEs), whose corresponding discrete forms are solved with the simplest Krylov subspace iterative method. Unlike the traditional boundary method, this method evaluates boundary or volume integrals encountered in BIEs by solving equivalent elasticity interface problem with Lame coefficients taken as constant one. By introducing an additional unknown, the equivalent elasticity interface problem can be rewritten a system like Stokes equations. The well-known marker and cell (MAC) finite difference scheme is used to discretize it, and the resulting linear system is solved by the CG method incorporated with the FFT-based fast solvers. Numerical results are given to demonstrated the efficiency and accuracy of the proposed methods.