We simulate the dilute attractive Fermi-Hubbard model in the unitarity regime using a diagrammatic determinant Monte Carlo algorithm with worm-type updates. We obtain the dependence of the critical temperature on the filling factor $\nu$ and, by extrapolating to $\nu \to 0$, determine the universal critical temperature of the continuum unitary Fermi gas in units of Fermi energy: $T_c/\epsilon_F=0.152(7)$. We also determine the thermodynamic functions and show how the Monte Carlo results can be used for accurate thermometry of a trapped unitary gas.