We present neutrino mass bounds using 900,000 luminous galaxies with photometric redshifts measured from Sloan Digital Sky Survey III Data Release 8. The galaxies have photometric redshifts between z = 0.45 and z = 0.65 and cover 10,000 deg(2), thus probing a volume of 3 h(-3) Gpc(3) and enabling tight constraints to be derived on the amount of dark matter in the form of massive neutrinos. A new bound on the sum of neutrino masses Sigma m nu < 0.27 eV, at the 95% confidence level (CL), is obtained after combining our sample of galaxies, which we call CMASS with Wilkinson Microwave Anisotropy Probe (WMAP) seven-year cosmic microwave background data and the most recent measurement of the Hubble parameter from the Hubble Space Telescope (HST). This constraint is obtained with a conservative multipole range of 30 < l < 200 in order to minimize nonlinearities, and a free bias parameter in each of the four redshift bins. We study the impact of assuming this linear galaxy bias model using mock catalogs and find that this model causes a small (similar to 1 sigma-1.5 sigma) bias in Omega(DM)h(2). For this reason, we also quote neutrino bounds based on a conservative galaxy bias model containing additional, shot-noise-like free parameters. In this conservative case, the bounds are significantly weakened, e. g., Sigma m(nu) < 0.38 eV (95% CL) for WMAP+HST+CMASS (l(max) = 200). We also study the dependence of the neutrino bound on the multipole range (l(max) = 150 versus l(max) = 200) and on which combination of data sets is included as a prior. The addition of supernova and/or baryon acoustic oscillation data does not significantly improve the neutrino mass bound once the HST prior is included. A companion paper describes the construction of the angular power spectra in detail and derives constraints on a general cosmological model, including the dark energy equation of state w and the spatial curvature Omega(K), while a second companion paper presents a measurement of the scale of baryon acoustic oscillations from the same data set. All three works are based on the catalog by Ross et al.