Anisotropic phonon transport is observed along different lattice directions in two-dimensional (2D) and layered materials. However, this effect vanishes in homogeneous, covalently bonded films, as the thickness is increased beyond few atomic layers. Here we establish a fundamental mechanism to induce anisotropic phonon transport in quasi-2D materials with in-plane isotropic symmetry. The anisotropy is engendered by the resonant modes of surface nanostructures that hybridize with membrane modes. Using atomistic lattice dynamics and classical molecular dynamics we demonstrate that the thermal conductivity of silicon membranes with surface nanofins is larger by approximately 100% parallel to the fins than in the perpendicular direction. The primary advantage of these configurations is that they would be technologically viable for implementation to existing and innovative material architectures. We anticipate that our results will open up alternative research directions to control phonons of technology-enabling nanomaterials for a broad range of applications, including thermal management in three-dimensional interconnected nanoelectronics, thermoelectric conversion to IR sensing.