Mergers of neutron-star/neutron-star binaries are strong sources of gravitational waves(1-3). They can also launch subrelativistic and mildly relativistic outflows(4-8) and are often assumed to be the sources of short gamma-ray bursts(9). An electromagnetic signature that persisted for weeks to months after the event would strengthen any future claim of a detection of gravitational waves(10). Here we present results of calculations showing that the interaction of mildly relativistic outflows with the surrounding medium produces radio flares with peak emission at 1.4 gigahertz that persist at detectable (submillijansky) levels for weeks, out to a redshift of 0.1. Slower subrelativistic outflows produce flares detectable for years at 150 megahertz, as well as at 1.4 gigahertz, from slightly shorter distances. The radio transient RT 19870422 (ref. 11) has the properties predicted by our model, and its most probable origin is the merger of a compact neutron-star/neutron-star binary. The lack of radio detections usually associated with short gamma-ray bursts does not constrain the radio transients that we discuss here (from mildly relativistic and subrelativistic outflows) because short gamma-ray burst redshifts are typically >0.1 and the appropriate timescales (longer than weeks) have not been sampled.