We present the results of our high-resolution spectroscopic follow-up of 42 planetary transit candidates in Carina from the OGLE survey. This follow-up has already allowed the discovery of three new transiting exoplanets, OGLE-TR-111, 113 and 132, presented in earlier Letters (Bouchy et al. 2004, Pont et al. 2004). Here we analyse the data for the remaining 39 candidates. The radial velocity data show that most of them are eclipsing binaries, in very varied configurations. Precise radial

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... y orbits were derived for 15 binaries, revealing 9 transits of small stars (generally M-dwarfs) in front of F-G dwarfs, 1 grazing equal-mass eclipsing binary, 4 triple and 1 quadruple systems. A remaining 14 systems appear binary, but the exact orbit is uncertain or was not determined. 2 objects do not show any radial velocity variations in phase with the transit signal, and 6 do not possess spectral lines strong enough for a reliable cross-correlation function to be measured. Among these last two categories, up to 6 objects are suspected false positives of the photometric transit detection. Finally 2 objects are unsolved cases that deserve further observations. Our study illustrates the wide variety of cases that can mimic photometric planetary transits, and the importance of spectroscopic follow-up. Multi-fiber capacities and an optimized follow-up strategy, which we present here, can help deal with the high number of candidates that are likely to turn up in the near future. (abridged)