The unitarity regime of the BCS-BEC crossover can be realized by diluting a system of two-component lattice fermions with an on-site attractive interaction. We perform a systematic-error-free finite-temperature simulation of this system by diagrammatic determinant Monte Carlo method. The critical temperature in units of Fermi energy is found to be Tc/εF=0.152(7). We also report the behavior of the thermodynamic functions, and discuss the issues of thermometry of ultracold Fermi gases.