Synchrotron-emitting, non-thermal filaments (NTFs) have been observed near the Galactic centre for nearly four decades, yet their physical origin remains unclear. Here we investigate the possibility that NTFs are produced by the destruction of molecular clouds by the gravitational potential of the Galactic centre. We show that this model predicts the formation of a filamentary structure with length on the order of tens to hundreds of pc, a highly ordered magnetic field along the axis of the filament, and conditions conducive to magnetic reconnection that result in particle acceleration. This model therefore yields the observed magnetic properties of NTFs and a population of relativistic electrons, without the need to appeal to a dipolar, ∼mG, Galactic magnetic field. As the clouds can be both completely or partially disrupted, this model provides a means of establishing the connection between filamentary structures and molecular clouds that is observed in some, but not all, cases.