Poly(lactic) acid nanocomposites filled with graphene nanoplatelets (GNPs) and multiwall carbon nanotubes (MWCNTs) are studied, varying the filler size, shape, and content within 1.5–12 wt.%. The effects of the intrinsic characteristics of nanofillers and structural organization of nanocomposites on mechanical, electrical, thermal, and electromagnetic properties enhancement are investigated. Three essential rheological parameters are identified, which determine rheology–structure–property relations in nanocomposites: the degree of dispersion, percolation threshold, and interfacial interactions. Above the percolation threshold, depending on the degree of dispersion, three structural organizations are observed in nanocomposites: homogeneous network (MWCNTs), segregated network (MWCNTs), and aggregated structure (GNPs). The rheological and structural parameters depend strongly on the type, size, shape, specific surface area, and functionalization of the fillers. Consequently, the homogeneous and segregated network structures resulted in a significant enhancement of tensile mechanical properties and a very low electrical percolation threshold, in contrast to the aggregated structure. The high filler density in the polymer and the low number of graphite walls in MWCNTs are found to be determinant for the remarkable shielding efficiency (close to 100%) of nanocomposites. Moreover, the 2D shaped GNPs predominantly enhance the thermal conductivity compared to the 1D shaped MWCNTs. The proposed essential structural parameters may be successfully used for the design of polymer nanocomposites with enhanced multifunctional properties for 3D printing applications.