This work shows the remarkable anisotropic nonlinear mechanical behavior of nanocomposites based on carbon nanotubes and poly(1,4-cis-isoprene), prepared by mechanical mixing, calendering, and compression molding. The dynamic mechanical moduli were measured as a function of the applied strain and along specific directions. The specimens revealed, at all filler contents, an orthotropic and transversally isotropic response, which means properties very similar inside the sheet plane and very different from those in the orthogonal direction. This response was correlated to the material structure by means of bright field transmission electron microscopy analysis, coupled with electron diffraction measurements, so to observe the nanofiller structuring through the specimen thickness. Preferential orientation of nanotubes and alternate areas containing large or low CNT amount were revealed. This work shows that energy dissipation is not isotropic in CNT-filled polymer nanocomposites and aims at giving a contribution for controlling such an important phenomenon.