Bulk polylactic acid (PLA)/multiwall carbon nanotube (MWCNT) composites were prepared and investigated in wide frequency ranges (20 Hz-1 MHz and 24-40 GHz). It was determined that the percolation threshold in bulk PLA/MWCNT composites is close to 0.2 vol.% MWCNT. However, the best microwave dielectric properties and absorption were observed in composites with 3.0-5.0 vol.% MWCNT. Therefore, for future investigations, we selected layered (laminate) polymeric structures with gradual changes in MWCNT concentration from 0.2 to 8.0 vol.% MWCNT. Two approaches to laminate structure designs were examined and compared: a five-layer composite and a nine-layer composite that included four pure PLA middle layers. The addition of MWCNT enhanced the elastic modulus by up to 1.4-fold and tensile strength by up to 1.2-fold, with the best performance achieved at 5.0 vol.% loading. High microwave shielding was observed for these layered PLA/MWCNT structures with a gradient change in MWCNT concentration (up to 26 dB in both transmission and absorption coefficients) in the broad frequency range (from 24 to 40 GHz). Obtained structures are highly anisotropic, and the absorption coefficient is 2-5 dB higher in the direction of MWCNT concentration increase; however, the transmission coefficient is the same in both directions. The properties of microwave absorption are mainly unaffected by the additional polymeric layers. The absorption of the layered structure is greater than the absorption of single-layer composites with an optimal MWCNT concentration of the same thickness. The proposed laminate structure design is promising in the field of efficient electromagnetic shielding.