Morphing an existing simulation mesh according to updated geometric parameters in the underlying computer-aided design model is a crucial technique within fully automatic design optimization. By avoiding costly automatic or even manual meshing, it enables the automatic and parallel generation and evaluation of new design variations, e.g., through finite element or computational fluid dynamics simulations. In this paper, we present a simple yet versatile method for high quality mesh morphing. Building upon triharmonic radial basis functions, our shape deformations minimize distortion and thereby implicitly preserve shape quality. Moreover, the same unified code can morph tetrahedral, hexahedral, or arbitrary polyhedral meshes. We compare our method to other recently proposed techniques and show that ours yields superior results in most cases. We analyze how to explicitly prevent inverted mesh elements by successively splitting the deformation into smaller steps. Finally, we investigate the performance of different linear solvers as well as the use of an incremental least squares solver for the sake of improved scalability.