© 2015 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society. Kinematical parametrizations of disc galaxies, employing emission line observations, are indispensable tools for studying the formation and evolution of galaxies. Future large-scale HI surveys will resolve the discs of many thousands of galaxies, allowing a statistical analysis of their disc and halo kinematics, mass distribution and dark matter content. Here, we present an automated procedure which fits tilted-ring models toHI data cubes of individual, well-resolved galaxies. The method builds on the 3D Tilted Ring Fitting Code (TIRIFIC) and is called Fully Automated TIRIFIC (FAT). To assess the accuracy of the code, we apply it to a set of 52 artificial galaxies and 25 real galaxies from the Local Volume HI Survey (LVHIS). Using LVHIS data, we compare our 3D modelling to the 2D modelling methods DISKFIT and ROTCUR. A conservative result is that FAT accurately models the kinematics and the morphologies of galaxies with an extent of eight beams across the major axis in the inclination range 20°-90° without the need for priors such as disc inclination. When comparing to 2D methods we find that velocity fields cannot be used to determine inclinations in galaxies that are marginally resolved.We conclude that with the current code tilted-ring models can be produced in a fully automated fashion. This will be essential for future HI surveys, with the Square Kilometre Array and its pathfinders, which will allow us to model the gas kinematics of many thousands of well-resolved galaxies. Performance studies of FAT close to our conservative limits, as well as the introduction of more parametrized models will open up the possibility to study even less resolved galaxies.