Photonic scattering materials, such as biological tissue and white paper, are made of randomly positioned nanoscale inhomogeneities in refractive index that lead to multiple scattering of light. Typically these materials, both naturally occurring or man-made, are formed through self assembly of the scattering inhomogeneities, making it extremely challenging to know the exact positions of these inhomogeneities, let alone control those. Here, the nanofabrication of photonic multiple-scattering media using direct laser writing with a deterministic design is reported on. These deterministic multiple-scattering media consist of submicron-thick polymer nanorods that are randomly oriented within a cubic volume. The total transmission of light is studied as a function of the number of rods and of the sample thickness to extract the scattering and transport mean free paths using radiative transfer theory. Such ability to fabricate photonic multiple-scattering media with deterministic and controllable properties opens up a myriad of opportunities for fundamental studies of light scattering, in particular, in the multiple-scattering regime and with strong anisotropy and for new applications in solid-state lighting and photovoltaics.