The Regular Hybrid Boundary Node Method (RHBNM) is developed in this paper for solving three-dimensional linear elasticity problems. Coupling modified functional with the Moving Least Squares (MLS) approximation, the RHBNM only requires discrete nodes constructed on the surface of a domain. Formulations and a general computer code of the RHBNM for 3D linear elasticity problems and the MLS interpolation on a generic surface have been developed. The RHBNM is formulated in terms of the domain and boundary variables. The domain variables are interpolated by classical fundamental solutions with the source points located outside the domain; and the boundary variables are interpolated by MLS approximation. The main idea is to retain the dimensionality advantages of the BNM, and localize the integration domain to a regular sub-domain, as in the MLBIE, such that no mesh is needed for integration. All integrals can be easily evaluated over regular shaped domains (in general, semi-sphere in the 3D problem) and their boundaries. Numerical examples for the solution of 3D elastostatic problems show that the high convergence rates with mesh refinement and the high accuracy with a small node number are achievable. The treatment of singularities and further integrations required for the computation of the unknown domain variables, as in the conventional BEM and BNM, can be avoided.