Oblivious routing has a long history in both the theory and practice of networking. In this dissertation, we initialize the formal study of oblivious routing in the context of reconfigurable networks, a new architecture that has recently come to the fore in data center networking, due to its increased energy efficiency and scaling potential. We focus on the tradeoffs between maximizing throughput and minimizing latency in this space. For every constant throughput rate, we characterize the minimum latency (up to a constant factor) achievable by an oblivious reconfigurable network design. The tradeoff curve turns out to be surprisingly subtle: it has an unexpected scalloped shape, reflecting the fact that routing becomes more costly when average path length is not an integer, since equalizing the path lengths is not achievable. We show that in order to guarantee the throughput value, Valiant load balancing is necessary, which lengthens routing paths by a factor of two. However, we also show that a strictly superior latency-throughput tradeoff is achievable when the throughput bound is relaxed to hold with high probability. The same improved tradeoff is also achievable with guaranteed throughput under time-stationary demands, provided the latency bound is relaxed to hold with high probability and that the network is allowed to be semi-oblivious, using an oblivious (randomized) connection schedule but demand-aware routing.