It is a common perception that the transport of a spin current in polycrystalline metal is isotropic and independent of the polarization direction, even though spin current is a tensorlike quantity and its polarization direction is a key variable. We demonstrate surprising anisotropic spin relaxation in mesoscopic polycrystalline Cu channels in nonlocal spin valves. For directions in the substrate plane, the spin-relaxation length is longer for spins parallel to the Cu channel than for spins perpendicular to it, by as much as 9% at 10 K. Spin-orbit effects on the surfaces of Cu channels can account for this anisotropic spin relaxation. The finding suggests novel tunability of spin current, not only by its polarization direction but also by electrostatic gating.