We present the first results from an on-going survey to characterize the circumgalactic medium (CGM) of the massive high-redshift galaxies detected as submillimeter galaxies (SMGs). We constructed a parent sample of 163 SMG-QSO pairs with separations less than $sim$36 arcsec by cross-matching far-infrared-selected galaxies from Herschel with spectroscopically confirmed QSOs. The Herschel sources were selected to match the properties of SMGs. We determined the sub-arcsecond positions of six Herschel sources with the Very Large Array and obtained secure redshift identification for three of those with near-infrared spectroscopy. The QSO sightlines probe transverse proper distances of 112, 157, and 198 kpc at foreground redshifts of 2.043, 2.515, and 2.184, respectively, which are comparable to the virial radius of the $sim10^{13}$ Msun halos expected to host SMGs. High-quality absorption-line spectroscopy of the QSOs reveals systematically strong HI Lyman-alpha absorption around all three SMGs, with rest-frame equivalent widths of $sim2-3$ AA. However, none of the three absorbers exhibits compelling evidence for optically thick HI gas or metal absorption, in contrast to the dominance of strong neutral absorbers in the CGM of luminous $z sim 2$ QSOs. The low covering factor of optically thick HI gas around SMGs tentatively indicates that SMGs may not have as prominent cool gas reservoirs in their halos as the co-eval QSOs and that they may inhabit less massive halos than previously thought.