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In this paper, we report the results of an X-ray monitoring campaign on the heavily obscured Seyfert galaxy, Markarian 3, carried out between the fall of 2014 and the spring of 2015 with NuSTAR, Suzaku and XMMNewton. The hard X-ray spectrum of Markarian 3 is variable on all the time-scales probed by our campaign, down to a few days. The observed continuum variability is due to an intrinsically variable primary continuum seen in transmission through a large, but still Compton-thin column density (N(sub H) approx. 0.8-1.1 x 10(exp 24)/sq cm). If arranged in a spherical-toroidal geometry, the Compton scattering matter has an opening angle approx. 66deg, and is seen at a grazing angle through its upper rim (inclination angle approx. 70deg). We report a possible occultation event during the 2014 campaign. If the torus is constituted by a system of clouds sharing the same column density, this event allows us to constrain their number (17 +/- 5) and individual column density, [approx. (4.9 +/- 1.5) x 10(exp 22)/ sq cm]. The comparison of IR and X-ray spectroscopic results with state-of-the art torus models suggests that at least two-thirds of the X-ray obscuring gas volume might be located within the dust sublimation radius. We report also the discovery of an ionized absorber, characterized by variable resonant absorption lines due to He- and H-like iron. This discovery lends support to the idea that moderate column density absorbers could be due to clouds evaporated at the outer surface of the torus, possibly accelerated by the radiation pressure due to the central AGN emission leaking through the patchy absorber.