The elastic constant tensor and its anisotropy are among the most critical mechanical properties, as they govern numerous mechanical phenomena and are prevalent in many natural materials. However, the efficient and accurate inverse design of metamaterials with desired elastic constants remains challenging, particularly for fully anisotropic elastic constants with low symmetries. Recent advances in artificial intelligence have opened new avenues to address this challenge. In this work, we propose a general framework that combines data-driven artificial neural networks with a gradient-based optimization algorithm to achieve high-precision inverse design of fully anisotropic elastic constants, exemplified using open cellular lattice Kelvin cells. First, an automatic parametric finite element method is introduced to calculate the elastic constants of any (distorted) Kelvin cells. Next, neural networks are developed to approximate the computationally costly finite element method, acting as the forward characterization function in the design process. Finally, an inverse design framework that integrates neural networks with a gradient-based optimization algorithm is proposed and validated. The successful design outcomes in practical examples, such as artificial bone implants and structures with unconventional Poisson's ratios, demonstrate the capability of our method to guide high-precision inverse design across various engineering applications.