Context. Observations of neutral hydrogen (H I) and molecular gas show that 50% of all nearby early-type galaxies contain some cold gas. Molecular gas is always found in small gas discs in the central region of the galaxy, while neutral hydrogen is often distributed in a low-column density disc or ring typically extending well beyond the stellar body. Dust is frequently found in early-type galaxies as well.Aims. The goal of our study is to understand the link between dust and cold gas in nearby early-type galaxies as a function of H I content.Methods. We analyse deep optical g-r images obtained with the MegaCam camera at the Canada-France-Hawaii Telescope for a sample of 21 H I-rich and 41 H I-poor early-type galaxies.Results. We find that all H I-rich galaxies contain dust seen as absorption. Moreover, in 57% of these H I-rich galaxies, the dust is distributed in a large-scale spiral pattern. Although the dust detection rate is relatively high in the H I-poor galaxies (similar to 59%), most of these systems exhibit simpler dust morphologies without any evidence of spiral structures. We find that the H I-rich galaxies possess more complex dust morphology extending to almost two times larger radii than H I-poor objects. We measured the dust content of the galaxies from the optical colour excess and find that H I-rich galaxies contain six times more dust (in mass) than H I-poor ones. In order to maintain the dust structures in the galaxies, continuous gas accretion is needed, and the substantial H I gas reservoirs in the outer regions of early-type galaxies can satisfy this need for a long time. We find that there is a good correspondence between the observed masses of the gas and dust, and it is also clear that dust is present in regions further than 3 R-eff.Conclusions. Our findings indicate an essential relation between the presence of cold gas and dust in early-type galaxies and offer a way to study the interstellar medium in more detail than what is possible with H I observations.