Reverberation mapping (RM) measurements of broad-line region (BLR) lags in z>0.3 quasars are important for directly measuring black hole masses in these distant objects, but so far there have been limited attempts and success given the practical difficulties of RM in this regime. Here we report preliminary results of 15 BLR lag measurements from the Sloan Digital Sky Survey Reverberation Mapping (SDSS-RM) project, a dedicated RM program with multi-object spectroscopy designed for RM over a wide redshift range. The lags are based on the 2014 spectroscopic light curves alone (32 epochs over 6 months) and focus on the Hbeta and MgII broad lines in the 100 lowest-redshift (z<0.8) quasars included in SDSS-RM; they represent a small subset of the lags that SDSS-RM (including 849 quasars to z~4.5) is expected to deliver. The reported preliminary lag measurements are for intermediate-luminosity quasars at 0.3<~z<0.8, including 9 Hbeta lags and 6 MgII lags, for the first time extending RM results to this redshift-luminosity regime and providing direct quasar black hole mass estimates over ~ half of cosmic time. The MgII lags also increase the number of known MgII lags by several-fold, and start to explore the utility of MgII for RM at high redshift. The location of these new lags at higher redshifts on the observed BLR size-luminosity relationship is statistically consistent with previous Hbeta results at z<0.3. However, an independent constraint on the relationship slope at z>0.3 is not yet possible due to the limitations in our current sample. Our results demonstrate the general feasibility and potential of multi-object RM for z>0.3 quasars.