Observations of gravitational waves from massive binary black-hole systems at cosmological distances can be used to search for a dependence of the speed of propagation of the waves on wavelength, and thereby to bound the mass of a hypothetical graviton. We study the effects of precessions of the spins of the black holes and of the orbital angular momentum on the process of parameter estimation based on the method of matched filtering of gravitational-wave signals vs theoretical template waveforms. For the proposed Laser Interferometer Space Antenna, we show that precessions, and the accompanying modulations of the gravitational waveforms, are effective in breaking degeneracies among the parameters being estimated, and effectively restore the achievable graviton-mass bounds to levels obtainable from binary inspirals without spin. For spinning, precessing binary black-hole systems of equal masses 10⁶M_⊙ at 3 Gpc, the lower bounds on the graviton Compton wavelength achievable are of the order of 5×10¹⁶km.