We report a novel, triple-layered composite composed of multi-walled nanotubes (MWNTs) dispersed in an epoxy/cotton fabric resistive layer that is sandwiched between two MWNT-dispersed E-glass fiber/epoxy dielectric absorbing layers. This structure efficiently absorbs microwave radiation over a frequency range of 2-18 GHz. This layered, composite structure is thin relative to previously developed, multi-layer systems with a resistive layer/low dielectric layer/resistive layer/low-dielectric layer structure. The paper-like resistive layer facilitates broadband microwave absorption by matching the impedance between the two absorbing layers. The E-glass fiber/epoxy laminate composites provide structural and mechanical support. The conductive MWNT filler controls dielectric losses through the formation of a network that allows high complex permittivity at low concentrations. Free-space measurements showed that the resulting triple-layered structure boasts an outstandingly broad bandwidth of reflection loss below -10 dB from 4.7 to 13.7 GHz. This new multi-layer structure offers a promising solution for attenuating electromagnetic waves over broad frequency ranges.