The extreme customisation and rapid prototyping capabilities of the 3D printing process allows the manufacture of low-cost and patient-specific radiotherapy phantoms for quality assurance purposes. However, the associated printing techniques and materials are experimentally limited and are yet to be quantified in terms of manufacturability. In addition to this, there lacks research in utilising naturally inspired structures, known as triply periodic minimal surfaces (TPMS), as a structural manufacturing basis for these phantoms, enabling material heterogeneity, which is a significant factor in attaining patient-specificity. We propose the use of Gyroid structures for radiotherapy phantom applications to investigate Gyroid-phantom manufacturability, the mathematical definition of Gyroids and their effects on hounsfield units (HU). The printed Gyroid phantoms were assessed for manufacturability using optical microscopy and micro-computed tomography (µCT), and material hounsfield-equivalence using standard medical CT. A mean HU range of -900 to -390 were achieved from the fabricated Gyroid phantoms with varying standard deviation. Compared to traditional printing infills such as grid and slit structures, the Gyroid phantoms were observed to produce isotropic HU and SD at varied scanning orientations, which optimizes CT attenuations in modulating the hounsfield-equivalence of printed structures. This study not only demonstrates the feasibility of manipulating the structural parameters of Gyroids in simulating tissue imaging attenuations but also opens significant research opportunities in fabricating patient-specific phantoms with added pathological features for end-to-end radiotherapy testing.