Barite and hematite are the most common heavy-weight minerals applied as aggregates in radiation shielding concrete (RSC). Therefore, to limit the cement consumption and reduce the CO2 emissions its production, the aim of this study is to use Egyptian barite and hematite minerals in their native status and evaluate their attenuation efficiency against fast neutrons and γ-rays. This was implemented through the measurement of their radiation attenuation against fast neutrons and γ-rays in the energy ranges of 0.80–11 and 0.40–8.30 MeV, respectively, employing a Pu-Be source and a stilbene scintillator. Theoretical calculations were prepared using the NXcom program to validate the fast neutron attenuation measurements. Furthermore, the implications of the physical, mineralogical, geochemical, and morphological characteristics of these heavy-weight minerals with respect to their attenuation efficiencies were considered. We found that barite has superior radiation attenuation efficiency for fast neutrons and γ-rays compared to hematite by 9.17 and 51% for fast neutrons and γ-rays, respectively. This was ascribed to the superior physical, mineralogical, geochemical, and morphological properties of the former relative to those of the latter. Furthermore, a satisfactory agreement between the experimental and theoretical results was achieved, with a deviation of 16 and 19.25% for the barite and hematite samples, respectively. Eventually, barite and hematite can be successful candidates for their use as sustainable alternatives to common RSC.