A shell-model method for calculating Gamow-Teller (GT) transition rates in heavy deformed odd-mass nuclei is presented. The method is developed within the framework of the projected shell model. To implement the computation requirement when many multi-quasiparticle configurations are included in the basis, a numerical advancement based on the Pfaffian formula is introduced. With this new many-body technique, it becomes feasible to perform state-by-state calculations for the GT nuclear matrix elements of beta-decay and electron-capture processes, including those at high excitation energies in heavy nuclei which are usually deformed. The first results, beta-decays of the well-deformed A = 153 neutron-rich nuclei, are shown as the example. The known log(ft) data corresponding to the B(GT(-)) decay rates of the ground state of Nd-153 to the low-lying states of Pm-153 are well described. It is further shown that the B(GT) distributions can have a strong dependence on the detailed microscopic structure of relevant states of both the parent and daughter nuclei.