We present intermediate-resolution (FWHM~1.5-2.6 Angstroms) optical spectra of 106 candidate optical counterparts of 77 X-ray sources detected in four pointed (t_exp>=8700 s) ROSAT PSPC observations of the rho Ophiuchi star-forming region and vicinity. Using the spectral types and equivalent widths of Hα and Liilambda670.8 nm obtained from our spectra, we applied spectroscopic criteria in order to classify our sample in different pre-main-sequence subtypes: `classical' T Tauri stars (CTTS); `weak' T Tauri stars (WTTS), and `post' T Tauri stars (PTTS). A total of 10 CTTS, 43 WTTS and 6 PTTS were found among the PSPC-selected stars. Our results more than double the number of pre-main-sequence stars spectroscopically identified in the rho Ophiuchi region. We considered regions with different molecular cloud properties: the central core and the outer ring of the rho Ophiuchi dark cloud (L1688); the `streamers' (L1709); the R7 clump; and the `smoke rings'. In the inner field of L1688, the ratio of WTTS over CTTS is ~1:1, significantly smaller than in the other regions (4:1 in the outer ring of L1688 and 5:1 in the smoke rings). The WTTS/CTTS ratio in the R7 field is the highest of our survey (10:1). We argue that this could be a result of the UV radiation from the nearby massive binary rho Oph AB, and/or of winds from the Upper Sco OB association, which might shorten the lifetime of the circumstellar discs of the low-mass stars. We find no PTTS in the inner field of our L1688 PSPC image, and only 3 PTTS in the outer ring despite the high sensitivity of our X-ray observations in this region. This result confirms that the central region of L1688 is extremely young (age<5 Myr), as suggested by near-infrared surveys. The presence of a small number of PTTS scattered around the rho Ophiuchi molecular clouds suggests that star formation may have been going on for 10 to 30 Myr, very slowly at first, but at a much higher rate for the last ~10 Myr. We provide rough estimates of the star formation rate for the main rho Ophiuchi molecular cloud complex.