Transverse momentum spectra of pions, kaons, protons and antiprotons from Au+Au collisions at sqrt(s_(NN)) = 62.4 GeV have been measured by the PHOBOS experiment at the Relativistic Heavy Ion Collider at Brookhaven National Laboratory. The identification of particles relies on three different methods: low momentum particles stopping in the first detector layers; the specific energy loss (dE/dx) in the silicon Spectrometer, and Time-of-Flight measurement. These methods cover the transverse momentum ranges 0.03-0.2, 0.2-1.0 and 0.5-3.0 GeV/c, respectively. Baryons are found to have substantially harder transverse momentum spectra than mesons. The pT region in which the proton to pion ratio reaches unity in central Au+Au collisions at sqrt(s_(NN)) = 62.4 GeV fits into a smooth trend as a function of collision energy. At low transverse momenta, the spectra exhibit a significant deviation from transverse mass scaling, and when the observed particle yields at very low pT are compared to extrapolations from higher pT, no significant excess is found. By comparing our results to Au+Au collisions at sqrt(s_(NN)) = 200 GeV, we conclude that the net proton yield at midrapidity is proportional to the number of participant nucleons in the collision.