  We report an incipient exploration of the Lense-Thirring precession effect in
a rotating \textit{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.
