This is the first of a series of papers in which the kinematics of disk galaxies over a range of scales is scrutinised by employing spectroscopy. A fundamental aspect of these studies is presented here: the new publicly available software tool TiRiFiC (http://www.astro.uni-bonn.de/(similar to) gjozsa/tirific.html) enables a direct fit of a "tilted-ring model" to spectroscopic data cubes. The algorithm generates model data cubes from the tilted-ring parametrisation of a rotating disk, which are automatically adjusted to reach an optimum fit via a chi-squared minimisation method to an observed data cube. The structure of the new software, the shortcomings of the previously available programs to produce a tilted-ring model, and the performance of TiRiFiC are discussed. Our method is less affected by the well-known problem of beam smearing that occurs when fitting to the velocity field. Since we fit many data points in a data cube simultaneously with our method, TiRiFiC is sensitive to very faint structures so can be used to derive tilted-ring models significantly extending in radius beyond those derived from a velocity field. The software is able to parametrise H I disks of galaxies that are intersected by the line-of-sight twice or more, i.e. if the disks are heavily warped, and/or with a significant shift of the projected centre of rotation, and/or if seen edge-on. Furthermore, our method delivers the surface-brightness profile of the examined galaxy in addition to the orientational parameters and the rotation curve. In order to derive kinematic and morphological models of disk galaxies, especially reliable rotation curves, a direct-fit method as implemented in our code should be the tool of choice.