We study the diffusive dynamics of a Brownian particle in the proximity of a flat surface under nonequilibrium conditions, which are created by an anisotropic thermal environment with different temperatures being active along distinct spatial directions. By presenting the exact time-dependent solution of the Fokker-Planck equation for this problem, we demonstrate that the interplay between anisotropic diffusion and hard-core interaction with the plain wall rectifies the thermal fluctuations and induces directed particle transport parallel to the surface, without any deterministic forces being applied in that direction. Based on current micromanipulation technologies, we suggest a concrete experimental setup to observe this novel noise-induced transport mechanism. We furthermore show that it is sensitive to particle characteristics, such that this setup can be used for sorting particles of different sizes.