We calculate the thermomechanical properties of alpha-iron, and in particular its isothermal and adiabatic elastic constants, using first-principles total-energy and lattice-dynamics calculations, minimizing the quasiharmonic vibrational free energy under finite strain deformations. Particular care is made in the fitting procedure for the static and temperature-dependent contributions to the free energy, in discussing error propagation for the two contributions separately, and in the verification and validation of pseudopotential and all-electron calculations. We find that the zero-temperature mechanical properties are sensitive to the details of the calculation strategy employed, and common semilocal exchange-correlation functionals provide only fair to good agreement with experimental elastic constants, while their temperature dependence is in excellent agreement with experiments in a wide range of temperature almost up to the Curie transition.