Context. The orbital obliquity of planets with respect to the rotational axis of their host stars is a relevant parameter for the characterization of the global architecture of planetary systems and a key observational constraint to discriminate between different scenarios proposed to explain the existence of close-in giant planets. Aims: In the framework of the GAPS project, we conduct an observational programme aimed at determinating the orbital obliquity of known transiting exoplanets. The targets are selected to probe the obliquity against a wide range of stellar and planetary physical parameters. Methods: We exploit high-precision radial velocity (RV) measurements, delivered by the HARPS-N spectrograph at the 3.6 m Telescopio Nazionale Galileo, to measure the Rossiter-McLaughlin (RM) effect in RV time-series bracketing planet transits, and to refine the orbital parameters determinations with out-of-transit RV data. We also analyse new transit light curves obtained with several 1-2 m class telescopes to better constrain the physical fundamental parameters of the planets and parent stars. Results: We report here on new transit spectroscopic observations for three very massive close-in giant planets: WASP-43 b, HAT-P-20 b and Qatar-2 b (Mp = 2.00, 7.22, 2.62 MJ; a = 0.015, 0.036, 0.022 AU, respectively) orbiting dwarf K-type stars with effective temperature well below 5000 K (Teff = 4500 ± 100, 4595 ± 45, 4640 ± 65 K respectively). These are the coolest stars (except for WASP-80) for which the RM effect has been observed so far. We find λ = 3.5 ± 6.8 deg for WASP-43 b and λ = -8.0 ± 6.9 deg for HAT-P-20 b, while for Qatar-2, our faintest target, the RM effect is only marginally detected, though our best-fit value λ = 15 ± 20 deg is in agreement with a previous determination. In combination with stellar rotational periods derived photometrically, we estimate the true spin-orbit angle, finding that WASP-43 b is aligned while the orbit of HAT-P-20 b presents a small but significant obliquity ( deg). By analyzing the CaII H&K chromospheric emission lines for HAT-P-20 and WASP-43, we find evidence for an enhanced level of stellar activity that is possibly induced by star-planet interactions. Based on observations collected at the Italian Telescopio Nazionale Galileo (TNG), operated on the island of La Palma by the Fundación Galileo Galilei of the Istituto Nazionale di Astrofisica (INAF) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the frame of the programme Global Architecture of Planetary Systems (GAPS).Also based on observations collected at the 0.82 m IAC80 Telescope, operated on the island of Tenerife by the Instituto de Astrofísica de Canarias in the Spanish Observatorio del Teide.