The aim of this paper is to (1) introduce an approach, called polytope sector-based synthesis (PSS), for synthesizing 2D or 3D microstructural architectures that exhibit a desired bulk-property directionality (e.g., isotropic, cubic, orthotropic, etc.), and (2) provide general analytical methods that can be used to rapidly optimize the geometric parameters of these architectures such that they achieve a desired combination of bulk thermal conductivity and thermal expansion properties. Although the methods introduced can be applied to general beam-based microstructural architectures, we demonstrate their utility in the context of an architecture that can be tuned to achieve a large range of extreme thermal expansion coefficients—positive, zero, and negative. The material-property-combination region that can be achieved by this architecture is determined within an Ashby-material-property plot of thermal expansion versus thermal conductivity using the analytical methods introduced. These methods are verified using finite-element analysis (FEA) and both 2D and 3D versions of the design have been fabricated using projection microstereolithography.