We report the results from recent observations of Sgr A* at short-/sub-millimeter wavelengths made with the partially finished Sub-Millimeter Array (SMA) on Mauna Kea. A total of 25 epochs of observations were carried out over the past 15 months in 2001 March to 2002 May. Noticeable variations in flux density at 1.3 mm were observed showing three ``flares''. The SMA observations suggest that Sgr A* highly increases towards submillimeter wavelengths during a flare suggesting the presence of a break wavelength in spectral index around 3 mm. A cross-correlation of the SMA data at 1 mm with the VLA data at 1 cm show a global delay of t_delay> 3 d, suggesting that sub-millimeter wavelengths tend to peak first. Only marginal day-to-day variations in flux density (2-3 sigma ) have been detected at 1.3 mm. No significant flares on a short time scale ( 1 hr) have been observed at 1.3 mm. We also failed to detect significant periodic signals at a level of 5% (3 sigma ) from Sgr A* in a periodic searching window ranging from 10 min to 2.5 hr. The flares observed at the wavelengths between short-centimeter and sub-millimeter might be a result of collective mass ejections associated with X-ray flares that originate from the inner region of the accretion disk near the supermassive black hole.