We explore opportunities for multi-messenger astronomy using gravitational waves (GWs) and prompt, transient low-frequency radio emission to study highly energetic astrophysical events. We review the literature on possible sources of correlated emission of gravitational waves and radio transients, highlighting proposed mechanisms that lead to a short-duration, high-flux radio pulse originating from the merger of two neutron stars or from a superconducting cosmic string cusp. We discuss the detection prospects for each of these mechanisms by low-frequency dipole array instruments such as LWA1, LOFAR and MWA. We find that a broad range of models may be tested by searching for radio pulses that, when de-dispersed, are temporally and spatially coincident with a LIGO/Virgo GW trigger within a $\usim 30$ second time window and $\usim 200 \mendash 500 \punits{deg}^{2}$ sky region. We consider various possible observing strategies and discuss their advantages and disadvantages. Uniquely, for low-frequency radio arrays, dispersion can delay the radio pulse until after low-latency GW data analysis has identified and reported an event candidate, enabling a \emph{prompt} radio signal to be captured by a deliberately targeted beam. If neutron star mergers do have detectable prompt radio emissions, a coincident search with the GW detector network and low-frequency radio arrays could increase the LIGO/Virgo effective search volume by up to a factor of $\usim 2$. For some models, we also map the parameter space that may be constrained by non-detections. ; Comment: 31 pages, 4 figures