The polarized mm/sub-mm radiation from Sgr A* is apparently produced by a Keplerian structure whose peak emission occurs within several Schwarzschild radii (rS\equiv 2GM/c2) of the black hole. The Chandra X-ray counterpart, if confirmed, is presumably the self-Comptonized component from this region. In this paper, we suggest that sub-mm timing observations could yield a signal corresponding to the period P0 of the marginally stable orbit, and therefore point directly to the black hole's spin a. Sgr A*'s mass is now known to be (2.6+/- 0.2)* 106 Mo (an unusually accurate value for supermassive black hole candidates), for which 2.7 min < P0 < 36 min, depending on the value of a and whether the Keplerian flow is prograde or retrograde. A Schwarzschild black hole (a=0) should have P0 20 min. The identification of the orbital frequency with the innermost stable circular orbit is made feasible by the transition from optically thick to thin emission at sub-mm wavelengths. With stratification in the emitter, the peak of the sub-mm bump in Sgr A*'s spectrum is thus produced at the smallest radius. We caution, however, that theoretical uncertainties in the structure of the emission region may still produce some ambiguity in the timing signal. Given that Sgr A*'s flux at nu 1 mm is several Jy, these periods should lie within the temporal-resolving capability of sub-mm telescopes using bolometric detectors. A determination of P0 should provide not only a value of a, but it should also define the angular momentum vector of the orbiting gas in relation to the black hole's spin axis. By analogy with low-mass X-ray binaries and Galactic black hole candidates, Sgr A* may also display quasi-periodic oscillations, which can reveal additional features in the geometry of the accreting gas. In addition, since the X-ray flux detected by Chandra appears to be the self-Comptonized mm to sub-mm component, these temporal fluctuations may also be evident in the X-ray signal.