Depending on the 3D-printing orientation, the anisotropy of the elastic characteristics of the Inconel 625 alloy generated with selective laser sintering from powders is investigated. The impact of the original powder combination and the following heat treatment (post-printing treatment) on the anisotropy of the alloy elastic characteristics is assessed. As demonstrated, the suggested treatments can lessen the anisotropy of the alloy elastic characteristics. Based on knowledge of the elastic constants of the single crystal and x-ray texture features, the results of a theoretical estimation of the elastic and shear moduli, Poisson’s ratio, and their anisotropy in the horizontal and vertical directions of 3D printing are provided. As demonstrated, the obtained theoretical values differ by 6–10% from the corresponding experimental values. The calculated stress–strain state can be more accurately calculated with the use of the estimated elastic characteristics and their anisotropy. The strategy for 3D-printing complicated components from Inconel 625 alloy may be made more effective.