As most arbitrarily shaped cloaks can be approximated by polyhedra and further divided into a series of tetrahedra, we propose in this paper a linear mapping approach to design cloaks with tetrahedron shapes (i.e. tetrahedral cloaks). Homogeneous material properties of the cloak are straightforwardly obtained from coordinates of typical points. Consequently, most arbitrarily shaped thermal cloaks can be designed using homogeneous anisotropic materials only. We construct two polyhedral cloaks and show numerically that they successfully thermally conceal objects after a certain lapse of time. We then demonstrate that cloaks with curved boundaries can also be obtained using our approach. It is further shown how geometrical parameters affect the material properties and cloaking performances. One can ﬂexibly tune the cloaking performances based on practical requirements and material availabilities. We analyze the effectiveness of a polygonal cloak composed of an alternation of homogeneous isotropic layers, and note that cloaking deteriorates at short times. We ﬁnally sketch an approach to realize 3D homogenized thermal metamaterials.