Metamaterial systems capable of exhibiting changes in thermal conductivities in response to an external control or input, as well as methods relating thereto. The metamaterial systems include first and second plates, and a metamaterial core between and thermally coupled to the first and second plates. The metamaterial core comprises a plurality of elements coupled to and contacting each other, with each of the elements being a pseudo-tetrahedron having surfaces that define surface-to-surface contacts with at least one other of the elements. A force is applied to the metamaterial core that increases contact pressures between the elements at the surface-to-surface contacts thereof and thereby increases thermal contact conductivities at the surface-to-surface contacts and increases a thermal conductivity of the metamaterial core.