We present the results of a multiplicity survey of 212 T Tauri stars in the Chamaeleon I and Taurus-Auriga star-forming regions, based on high-resolution spectra from the Magellan Clay 6.5 m telescope. From these data, we achieved a typical radial velocity (RV) precision of similar to 80 m s(-1) with slower rotators yielding better precision, in general. For 174 of these stars, we obtained multi-epoch data with sufficient time baselines to identify binaries based on RV variations. We identified eight close binaries and four close triples, of which three and two, respectively, are new discoveries. The spectroscopic multiplicity fractions we find for Chamaeleon I (7%) and Taurus-Auriga (6%) are similar to each other, and to the results of field star surveys in the same mass and period regime. However, unlike the results from imaging surveys, the frequency of systems with close companions in our sample is not seen to depend on primary mass. Additionally, we do not find a strong correlation between accretion and close multiplicity. This implies that close companions are not likely the main source of the accretion shut down observed in weak-lined T Tauri stars. Our results also suggest that sufficient RV precision can be achieved for at least a subset of slowly rotating young stars to search for hot Jupiter planets.