This article presents a new approach for inverse identification of all elastic constants of a 3D generally anisotropic solid with arbitrary geometry via measured strain data. To eradicate the nonlinear inequality constraints posed on the elastic constants, the problem is first transformed to an unconstrained one by the Cholesky factorization theorem. The cost function is defined by the Tikhonov regularization method, and the inverse problem is solved using the damped Gauss-Newton technique, where a meshless method is employed for the direct and sensitivity analyses. To demonstrate the effectiveness of the proposed approach, several examples are presented in the end, where all experimental data are numerically simulated. Analyses of these examples show that all twenty-one elastic constants of an example material can be correctly identified even when measurement errors are relatively large and initial guesses are far from exact values.