High-voltage electrical fields and low production rate limit electrospinning, the electrical charging of polymer liquids, as a means of nanofiber fabrication. Here, we show a facile method of fabrication of aligned three-dimensional nanofiber structures by utilizing high-speed, rotating polymer solution jets to extrude fibers. Termed rotary jet-spinning, fiber morphology, diameter, and web porosity can be controlled by varying nozzle geometry, rotation speed, and polymer solution properties. We demonstrate the utility of this technique for tissue engineering by building anisotropic arrays of biodegradable polymer fibers and seeding the constructs with neonatal rat ventricular cardiomyocytes. The myocytes used the aligned fibers to orient their contractile cytoskeleton and to self-organize into a beating, multicellular tissue that mimics the laminar, anisotropic architecture of the heart muscle. This technique may prove advantageous for building uniaxially aligned nanofiber structures for polymers which are not amenable to fabrication by electrospinning.