Architectural designs for DNA nanostructures typically fall within one of two broad categories: tile-based designs (assembled from chemically synthesized oligonucleotides) and origami designs (woven structures employing a biological scaffold strand and synthetic staple strands). Both previous designs typically contain many Holliday-type multi-arm junctions. Here we describe the design, implementation, and testing of a unique architectural strategy incorporating some aspects of each of the two previous design categories but without multi-arm junction motifs. Goals for the new design were to use only chemically synthesized DNA, to minimize the number of component strands, and to mimic the back-and-forth, woven strand routing of the origami architectures. The resulting architectural strategy employs "weave tiles" formed from only two oligonucleotides as basic building blocks, thus decreasing the burden of matching multiple strand stoichiometries compared to previous tile-based architectures and resulting in a structurally flexible tile. As an example application, we have shown that the four-helix weave tile can be used to increase the anticoagulant activity of thrombin-binding aptamers in vitro.