The Poisson's ratio νij = -ε/ε, where ε and ε (i,j = x, y, z) are applied and resulting strain, respectively, are computed from first-principles for Si with an array of cylindrical, nanometer-sized pores aligned in the z direction (nanoporous Si, or np-Si). Through density functional theory calculations, it is demonstrated that the periodic arrangement of pores introduces strong anisotropy in the Poisson's ratio of np-Si: while νyz remains close to the Poisson's ratio of the bulk, νzx and νxy exhibit an increase and a sharp decrease from the bulk value, respectively, as the volume fraction of pores (ϕ) becomes large. It is shown that the characteristic dependence of the Poisson's ratio on ϕ originates from the difference in the actual stress on np-Si, which is caused by the dissimilar surface geometry. Unlike random porous materials, this finding signifies the importance of structural details in determining the mechanical response of ordered systems at a nanoscale.