A fully relativistic study of gravitational radiation from bodies in circular equatorial orbits around the massive black hole at the Galactic Center, Sgr A*, has been performed, taking into account the Roche limit induced by tidal forces in the Kerr metric. The signal-to-noise ratio for LISA, as well as the time spent in LISA band, has been computed. It is found that brown dwarfs, main sequence stars and compact objects are all detectable in one year of LISA data with a signal-to-noise ratio above 10 during at least 10^5 yr in the slow inspiral towards either the innermost stable circular orbit (compact objects) or the Roche limit (main-sequence stars and brown dwarfs). The longest times in-band, of the order of 10^6 yr, are achieved for primordial black holes (mass < 10^{-3} solar mass), as well as for brown dwarfs, just followed by white dwarfs and low mass main-sequence stars.