We report results from a high-resolution optical spectroscopic survey aimed to search for nearby young associations and young stars among optical counterparts of ROSAT All-Sky Survey (<IX/10> and <IX/29>, X-ray sources in the Southern Hemisphere. We selected 1953 late-type (B-V>=0.6), potentially young, optical counterparts out of a total of 9574 1RXS sources for follow-up observations. At least one high-resolution spectrum was obtained for each of 1511 targets. This paper is the first in a series presenting the results of the SACY survey. Here we describe our sample and our observations. We describe a convergence method in the (UVW) velocity space to find associations. As an example, we discuss the validity of this method in the framework of the Beta Pic Association.