The high-density integration in information technology fuels the research on functional 3D nanodevices. Particularly ferromagnets promise multifunctional 3D devices for nonvolatile data storage, high-speed data processing, and non-charge-based logic operations via spintronics and magnonics concepts. However, 3D nanofabrication of ferromagnets is extremely challenging. In this work, an additive manufacturing methodology is reported, and unprecedented 3D ferromagnetic nanonetworks with a woodpile-structure unit cell are fabricated. The collective spin dynamics (magnons) at frequencies up to 25 GHz are investigated by Brillouin Light Scattering (BLS) microscopy and micromagnetic simulations. A clear discrepancy of about 10 GHz is found between the bulk and surface modes, which are engineered by different unit cell sizes in the Ni-based nanonetworks. The angle- and spatially-dependent modes demonstrate opportunities for multi-frequency signal processing in 3D circuits via magnons. The developed synthesis route will allow one to create 3D magnonic crystals with chiral unit cells, which are a prerequisite toward surface modes with topologically protected properties.