Recent observations with the Hinode Solar Optical Telescope display an active region prominence whose fine threads oscillate in the vertical direction as they move along a path parallel to the photosphere. A seismological analysis of this event is carried out by taking advantage of the small radius of these structures compared to the total length of magnetic field lines, i.e., by using the thin-tube approximation. This analysis reveals that the oscillatory period is only slightly modified by the existence of the flow and that the difference between the period of a flowing thread and a static one is below the error bars of these observations. Moreover, although it is not possible to obtain values of the physical parameters, a lower bound for the Alfvén speed (ranging between 120 and 350 km s-1) is obtained for each of the threads. Such Alfvén speeds agree with the intense magnetic fields and large densities usually found in active region prominences.