We present a novel network-on-chip-based architecture for future programmable chips (FPGAs). A key challenge for FPGA design is supporting numerous highly variable design instances with good performance and low cost. Our architecture minimizes the cost of supporting a wide range of design instances with given throughput requirements by balancing the amount of efficient hard-coded NoC infrastructure and the allocation of "soft" networking resources at configuration time. Although traffic patterns are design-specific, the physical link infrastructure is a performance bottleneck, and hence should be hard-coded. It is therefore important to employ routing schemes that allow for high flexibility to efficiently accommodate different traffic patterns during configuration. We examine the required capacity allocation for supporting a collection of typical traffic patterns on such chips under a number of routing schemes. We propose a new routing scheme, Weighted Ordered Toggle (WOT), and show that it allows high design flexibility with low infrastructure cost. Moreover, WOT utilizes simple, small-area, on-chip routers, and has low memory demands.