We investigate four young, but non-accreting, very low mass stars in Orion, which show irregular eclipses by circumstellar dust. The eclipses are not recurring periodically, are variable in depth, lack a flat bottom, and their duration is comparable to the typical timescale between eclipses. The dimming is associated with reddening consistent with dust extinction. Taken together this implies the presence of rings around these four stars, with radii ranging from 0.01 to 40 AU, comprised of optically thin dust clouds. The stars also show IR excess indicating the presence of evolved circumstellar disks, with orders of magnitude more material than needed for the eclipses. However, the rings need to cover an opening angle of about 20 degrees to explain how common these variable stars are in the coeval population in the same region, which is more extended than a typical disk. Thus, we propose that the rings may not be part of the disks, but instead separate structures with larger scale heights. To be sustained over years, the rings need to be replenished by dust from the disk or gravitationally bound to an object (e.g., planets or planetesimals). These four stars belong to a growing and diverse class of post-T Tauri stars with dips or eclipses in their lightcurves. Dusty rings with scale heights exceeding those of disks may be a common phenomenon at stellar ages between 5 and 10 Myr, in the transition from accretion disks to debris disks. These structures could be caused by migrating planets and may be signposts for the presence of young planetary systems.
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