In this paper, we present the 3D-1D asymptotic analysis of the Dirichlet spectral problem associated with an elliptic operator with axial periodic heterogeneities. We extend to the 3D-1D case previous 3D-2D results (see [R. Ferreira, M.L. Mascarenhas, and A. Piatnitski. ESAIM Control Optim. Calc. Var. 18, 2 (2012), 427–451]) and we analyze the special case where the scale of thickness is much smaller than the scale of the heterogeneities and the planar coefficient has a unique global minimum in the periodic cell. These results are of great relevance in the comprehension of the wave propagation in nanowires showing axial heterogeneities (see [L.J. Lauhon, M.S. Gudiksen, and C.M. Lieber, Philos. Trans. R. Soc. Lond. Ser. A-Math. Phys. Eng. Sci. 362, 1819 (2004), 1247–1260]).