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1/2 network capacity is often believed to be the limit of worst-case throughput for three-dimension (3D) mesh networks. However, this paper provides a new worst-case throughput bound, which is higher than 1/2 network capacity, for odd radix 3D mesh networks. In addition, we propose a routing algorithm called uniform solo-minimal (USM) routing that can achieve this new worst-case throughput bound in odd radix mesh networks. For the even radix case, we prove that USM achieves the optimal worst-case throughput, namely, half of network capacity. USM considers all routing paths with at most one dimensional minimal-distance routing and distributes the traffic loads uniformly to other two left dimensions. Theoretical analysis and simulation results show that USM outperforms existing routing algorithms in worst-case throughput. Moreover, USM achieves good average-throughput and performs very well under different traffic matrices at the expense of (5/3)× minimal average hop count.