Terahertz radiation has uses in applications ranging from security to medicine. However, sensitive room-temperature detection of terahertz radiation is notoriously difficult. The hot-electron photothermoelectric effect in graphene is a promising detection mechanism; photoexcited carriers rapidly thermalize due to strong electron-electron interactions, but lose energy to the lattice more slowly. The electron temperature gradient drives electron diffusion, and asymmetry due to local gating or dissimilar contact metals produces a net current via the thermoelectric effect. Here, we demonstrate a graphene thermoelectric terahertz photodetector with sensitivity exceeding 10VW-1 (700VW-1) at room temperature and noise-equivalent power less than 1,100pWHz-1/2 (20pWHz-1/2), referenced to the incident (absorbed) power. This implies a performance that is competitive with the best room-temperature terahertz detectors for an optimally coupled device, and time-resolved measurements indicate that our graphene detector is eight to nine orders of magnitude faster than those. A simple model of the response, including contact asymmetries (resistance, work function and Fermi-energy pinning) reproduces the qualitative features of the data, and indicates that orders-of-magnitude sensitivity improvements are possible.