We report an incipient exploration of the Lense‐Thirring precession effect in a rotating acoustic analogue black hole spacetime. An exact formula is deduced for the precession frequency of a gyroscope due to inertial frame dragging, close to the ergosphere of a ‘Draining Bathtub’ acoustic spacetime which has been studied extensively for acoustic Hawking radiation of phonons and also for ‘superresonance’. The formula is verified by embedding the two dimensional spatial (acoustic) geometry into a three dimensional one where the similarity with standard Lense‐Thirring precession results within a strong gravity framework is well known. Prospects of experimental detection of this new ‘fixed‐metric’ effect in acoustic geometries, are briefly discussed.