Increasing performance demands on photodetectors and solar cells require the development of entirely new materials and technological approaches. We report on the fabrication and optoelectronic characterization of a photodetector based on optically-thick films of dense, aligned, and macroscopically long single-wall carbon nanotubes. The photodetector exhibits broadband response from the visible to the mid-infrared under global illumination, with a response time less than 32 mu s. Scanning photocurrent microscopy indicates that the signal originates at the contact edges, with an amplitude and width that can be tailored by choosing different contact metals. A theoretical model demonstrates the photothermoelectric origin of the photoresponse due to gradients in the nanotube Seebeck coefficient near the contacts. The experimental and theoretical results open a new path for the realization of optoelectronic devices based on three-dimensionally organized nanotubes.