In our work, we employ the resonant electromagnetic properties of III-V semiconductor nanowires to design building blocks for nonlinear all-dielectric metamaterials and devices. Contrary to widely used Si and Ge nanostructures, III-V materials, such as GaAs or AlGaAs, have a direct band gap and non-centrosymmetric crystal structure, which makes them promising for the development of nonlinear metamaterials. We developed an innovative approach to fabricate disk and rod nanoantennas by slicing bottom-up grown nanowires using a focused ion beam milling (FIB). The proposed method allows to significantly decrease the influence of the substrate on the electromagnetic field distribution inside the nanoantenna and it opens the possibility to use any substrate regardless of the nanostructure fabrication process. With this technique, we study the influence of geometry, design and crystal structure on the characteristics of all-dielectric nanoantennas. It offers unique opportunities to fabricate high-quality structures with variable radii, longitudinal heterostructures with lattice-mismatched materials, and structures with different refractive indexes and crystal phases that are not available in bulk materials.