This paper describes the pioneering application of the meshfree solution structure method (SSM) to computer simulation of natural vibrations of 3D mechanical parts and structures. Using several carefully chosen examples, we investigate the accuracy and convergence of the computed natural frequencies. The salient feature of our approach is exact treatment of the prescribed boundary conditions that are enforced using approximate distance functions that vanish on the boundaries of a geometric model. Ability to use spatial meshes that do not necessarily conform to the shape of the geometric model makes it possible to eliminate or substantially simplify the finite element meshing. This defines unprecedented geometric flexibility of the SSM as well as the complete automation of the solution procedure.