Some active galactic nuclei (AGN) are surrounded by extended emission-line regions (EELRs), which trace both the illumination pattern of escaping radiation and its history over the light travel time from the AGN to the gas. From a new set of such EELRs, we present evidence that the AGN in many Seyfert galaxies undergo luminous episodes 0.2–2 years in duration. Motivated by the discovery of the spectacular nebula known as Hanny’s Voorwerp, ionized by a powerful AGN which has apparently faded dramatically within years, Galaxy Zoo volunteers have carried out both targeted and serendipitous searches for similar emission-line clouds around low-redshift galaxies. We present the resulting list of candidates and describe spectroscopy identifying 19 galaxies with AGN-ionized regions at projected radii kpc. This search recovered known EELRs (such as Mrk 78, Mrk 266 and NGC 5252) and identified additional previously unknown cases, one with detected emission to kpc. One new Sy 2 was identified. At least 14/19 are in interacting or merging systems, suggesting that tidal tails are a prime source of distant gas out of the galaxy plane to be ionized by an AGN. We see a mix of one- and two-sided structures, with observed cone angles from 23 to 112. We consider the energy balance in the ionized clouds, with lower and upper bounds on ionizing luminosity from recombination and ionization-parameter arguments, and estimate the luminosity of the core from the far-infrared data. The implied ratio of ionizing radiation seen by the clouds to that emitted by the nucleus, on the assumption of a non-variable nuclear source, ranges from 0.02 to ; 7/19 exceed unity. Small values fit well with a heavily obscured AGN in which only a small fraction of the ionizing output escapes to be traced by surrounding gas. However, large values may require that the AGN has faded over tens of thousands of years, giving us several examples of systems in which such dramatic long-period variation has occurred; this is the only current technique for addressing these time-scales in AGN history. The relative numbers of faded and non-faded objects we infer, and the projected extents of the ionized regions, give our estimate (0.2–2 years) for the length of individual bright phases.