Noble-metal nanocrystals with well-defined and controllable morphologies are of great importance to applications in catalysis, plasmonics, and surface-enhanced spectroscopy. Many synthetic approaches have been demonstrated for controlling the growth habit and thus morphology of metal nanocrystals, but most of them are based on a thermodynamic approach, including the use of a capping agent. While thermodynamic control has shown its power in generating nanocrystals with a myriad of different morphologies, it is ultimately limited by the obligation to minimize the surface energy of a system. As a result, it is impractical to use thermodynamic control to generate nanocrystals having high-energy facets and/or a negative curvature. Using rhodium as an example, here we demonstrate a general method based on kinetic control with a syringe pump that can be potentially extended to other noble metals and even other solid materials. For the first time, we were able to produce concave nanocubes with a large fraction of {110} facets and octapods with a cubic symmetry in high yields by simply controlling the injection rate at which the precursor was added into the reaction solution. The concave nanocubes with {110} facets and a unique cavity structure on the surface are important for a variety of applications.