This is the author accepted manuscript. The final version is available from AIP via http://dx.doi.org/10.1063/1.4928599 ; We apply Thermostatted Ring Polymer Molecular Dynamics (TRPMD), a recently-proposed approximate quantum dynamics method, to the computation of thermal reaction rates. Its short-time Transition-State Theory (TST) limit is identical to rigorous Quantum Transition-State Theory, and we find that its long-time limit is independent of the location of the dividing surface. TRPMD rate theory is then applied to one-dimensional model systems, the atom-diatom bimolecular reactions H+H_2, D+MuH and F+H_2, and the prototypical polyatomic reaction H+CH_4. Above the crossover temperature, the TRPMD rate is virtually invariant to the strength of the friction applied to the internal ring-polymer normal modes, and beneath the crossover temperature the TRPMD rate generally decreases with increasing friction, in agreement with the predictions of Kramers theory. We therefore find that TRPMD is approximately equal to, or less accurate than, Ring Polymer Molecular Dynamics (RPMD) for symmetric reactions, and for certain asymmetric systems and friction parameters closer to the quantum result, providing a basis for further assessment of the accuracy of this method. ; TJHH acknowledges a Research Fellowship from Jesus College, Cambridge, and helpful comments on the manuscript from Stuart Althorpe. YVS acknowledges support via the Newton International Alumni Scheme from the Royal Society. YVS also thanks the European Regional Development Fund and the Republic of Cyprus for support through the Research Promotion Foundation (Project Cy-Tera ??? ???????/??????/0308/31).