Selective laser melting (SLM) technology can rapidly form metal parts with complex geometry, but the internal microstructure and material properties are significantly different from those of traditionally manufactured parts, such as obvious anisotropy and inhomogeneity. This paper proposed a tri-mode ultrasonic system to nondestructively measure the anisotropic elastic constants of 316L stainless steel prepared by SLM. Firstly, the quantitative relationship between ultrasonic wave velocities and anisotropic elastic constants was derived for elliptical anisotropic materials. The anisotropic elastic constants and Thomsen anisotropic parameters can be determined by measuring four ultrasonic wave velocities V_11, V_33, V_12, and V_13. Therefore, a tri-mode ultrasonic transducer was designed and fabricated to build the elastic constant ultrasonic measurement system. Then, the elastic constants and Thomsen anisotropic parameters of SLM samples were measured, and the key process parameters’ influence was discussed. The standard static tensile test validated the result of the ultrasonic test, and the maximum relative error was 6.94%. This study will provide a novel nondestructive measurement method and system of anisotropic elastic constants to ensure internal quality and improve the additive manufacturing process.