The nature of additive manufacturing (AM) processes prescribes direction-dependent properties of the final parts. The degree of material anisotropy is highly dependent on the process parameters and the machine setup which complicates the design of AM parts. A basic problem in the design and quality control of parts and components manufactured by the AM processes is the evaluation of the resulting elastic properties, specifically along the principal directions. In a destructive testing approach, many specimens in the principal directions are normally required to determine the elastic properties of a material. However, an alternative low-cost method based on the ultrasonic wave propagation velocities can also be used for this purpose. In this article, an ultrasonic-based method for the determination of the elastic constants of the Inconel 625 (IN625) material as manufactured via the laser powder-bed fusion process (L-PBF) is presented. Several specimens are fabricated with various process parameters such as laser power, scan speed, and hatch spacing, and nondestructively tested. The material elastic constants are then determined by measuring the ultrasonic wave velocities within the specimen. The results are verified qualitatively with the published results and destructive tensile tests. The obtained results showed a good correlation indicating the effectiveness of the proposed method for the determination of elastic constants of additively manufactured IN625 material.