We report the results of our campaign to follow up spectroscopically several candidate extrasolar transiting planets from the OGLE-III survey in the direction of the Galactic center, announced in 2001. All of these objects present shallow and periodic dips in brightness that may be due to planetary companions. Our Keck I/High Resolution Echelle Spectrometer (HIRES) observations have revealed two interesting cases (OGLE-TR-10 with a period of 3.1 days and OGLE-TR-58 with a period of 4.3 days) that show no radial velocity variations at the level of 100-200 m s^-1. If orbited by companions, their masses would be similar to Jupiter. With the information in hand (including the light curves) we are not able to rule out that these candidates are instead the result of contamination from an eclipsing binary in the same line of sight (a "blend"). We also discuss the case of OGLE-TR-56, which was recently reported by Konacki et al. to have a Jupiter-size companion, on the basis of an earlier analysis of our data and we present supporting information. Two other candidates, OGLE-TR-3 and OGLE-TR-33, show clear evidence that they are blends. We describe tests carried out to characterize the stability of the HIRES spectrograph and its impact on the determination of precise velocities for faint stars (V ≥ 15 mag) using exposures of a thorium-argon lamp as the wavelength reference. Systematic effects are at the level of 100 m s^-1 or smaller and tend to dominate the total error budget. We also evaluate the precision attainable using the iodine gas absorption cell as an alternative fiducial, and we propose a simplified version of the standard procedure employed for high-precision Doppler planet searches that is very promising. Results from both this method and the classical ThAr technique show the feasibility of spectroscopic follow-up for faint targets in the range V = 14-17. We point out also that the high incidence of contamination from blends and other false positives in the OGLE-III survey already mentioned by Konacki et al. is potentially a serious concern not only for other similar surveys of relatively faint stars but also for future space missions to search for transits due to extrasolar planets as small as the Earth.