This work explores connections between FFT-based computational micromechanics and a homogenization approach based on the finite Radon transform introduced by Derraz and co-workers. We revisit periodic homogenization from a Radon point of view and derive the multidimensional Radon series representation of a periodic function from scratch. We introduce a general discretization framework based on trigonometric polynomials which permits to represent both the classical Moulinec-Suquet discretization and the finite Radon approach by Derraz et al. We use this framework to introduce a novel Radon framework which combines the advantages of both the Moulinec-Suquet discretization and the Radon approach, i.e., we construct a discretization which is both convergent under grid refinement and is able to represent certain non-axis aligned laminates exactly. We present our findings in the context of small-strain mechanics, extending the work of Derraz et al. that was restricted to conductivity and report on a number of interesting numerical examples.