We report Atacama Large Millimeter/submillimeter Array (ALMA) observations of CO(3-2) emission in a sample of seven Seyfert/LINER galaxies at the unprecedented spatial resolution of 0 ''.1 = 4-9 pc. Our aim is to explore the close environment of active galactic nuclei (AGN), and the dynamical structures leading to their fueling, through the morphology and kinematics of the gas inside the sphere of influence of the black hole. The selected galaxies host low-luminosity AGN and have a wide range of activity types (Seyferts 1 to 2, LINERs), and barred or ringed morphologies. The observed maps reveal the existence of circumnuclear disk structures, defined by their morphology and decoupled kinematics, in most of the sample. We call these structures molecular tori, even though they often appear as disks without holes in the center. They have varying orientations along the line of sight, unaligned with the host galaxy orientation. The radius of the tori ranges from 6 to 27 pc, and their mass from 0.7 x10(7) to 3.9 x 10(7 )M(circle dot). The most edge-on orientations of the torus correspond to obscured Seyferts. In only one case (NGC 1365), the AGN is centered on the central gas hole of the torus. On a larger scale, the gas is always piled up in a few resonant rings 100 pc in scale that play the role of a reservoir to fuel the nucleus. In some cases, a trailing spiral is observed inside the ring, providing evidence for feeding processes. More frequently, the torus and the AGN are slightly off-centered with respect to the bar-resonant ring position, implying that the black hole is wandering by a few 10 pc amplitude around the center of mass of the galaxy. Our spatial resolution allows us to measure gas velocities inside the sphere of influence of the central black holes. By fitting the observations with different simulated cubes, varying the torus inclination and the black hole mass, it is possible to estimate the mass of the central black hole, which is in general difficult for such late-type galaxies, with only a pseudo-bulge. In some cases, AGN feedback is revealed through a molecular outflow, which will be studied in detail in a subsequent article.