We investigate the possible bounds which could be placed on alternative theories of gravity using gravitational wave detection from inspiralling compact binaries with the proposed LISA space interferometer. Specifically, we estimate lower bounds on the coupling parameter ω of scalar–tensor theories of the Brans–Dicke type and on the Compton wavelength of the graviton λ_g in hypothetical massive graviton theories. In these theories, modifications of the gravitational radiation damping formulae or of the propagation of the waves translate into a change in the phase evolution of the observed gravitational waveform. We obtain the bounds through the technique of matched filtering, employing the LISA sensitivity curve generator (SCG), available online. For a non-spinning neutron star on a quasi-circular inspiral into a non-spinning 10³M black hole in the Virgo Cluster, in a two-year integration, we find a lower bound ω > 3 × 10⁵. For lower-mass black holes, the bound could be as large as 2 × 10⁶. The bound is independent of LISA arm length, but is inversely proportional to the LISA position noise error, under the assumption that position error noise dominates laser shot noise. Lower bounds on the graviton Compton wavelength ranging from 10¹⁵ km to 5 × 10¹⁶ km can be obtained from one-year observations of massive binary black-hole inspirals at cosmological distances (3 Gpc) for masses ranging from 10⁴ to 10⁷M. For the highest-mass systems (10⁷M), the bound is proportional to (LISA arm length)^1/2 and to (LISA acceleration noise)^−1/2. For the others, the bound is independent of these parameters because of the dominance of white-dwarf confusion noise in the relevant part of the frequency spectrum. These bounds improve and extend earlier work which used analytic formulae for the noise curves.