The problem of multi-object tracking (MOT) consists in detecting and tracking all the objects in a video sequence while keeping a unique identifier for each object. It is a challenging and fundamental problem for robotics. In precision agriculture the challenge of achieving a satisfactory solution is amplified by extreme camera motion, sudden illumination changes, and strong occlusions. Most modern trackers rely on the appearance of objects rather than motion for association, which can be ineffective when most targets are static objects with the same appearance, as in the agricultural case. To this end, on the trail of SORT [5], we propose AgriSORT, a simple, online, real-time tracking-by-detection pipeline for precision agriculture based only on motion information that allows for accurate and fast propagation of tracks between frames. The main focuses of AgriSORT are efficiency, flexibility, minimal dependencies, and ease of deployment on robotic platforms. We test the proposed pipeline on a novel MOT benchmark specifically tailored for the agricultural context, based on video sequences taken in a table grape vineyard, particularly challenging due to strong self-similarity and density of the instances. Both the code and the dataset are available for future comparisons.