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This letter presents a new oblivious routing algorithm for 3D mesh networks called Randomized Partially- Minimal (RPM) routing that provably achieves optimal worstcase throughput for 3D meshes when the network radix k is even and within a factor of 1/k2 of optimal when k is odd. Although this optimality result has been achieved with the minimal routing algorithm O1TURN  for the 2D case, the worst-case throughput of O1TURN degrades tremendously in higher dimensions. Other existing routing algorithms suffer from either poor worst-case throughput (DOR , ROMM ) or poor latency (VAL ). RPM on the other hand achieves near optimal worst-case and good average-case throughput as well as good latency performance.