Using speckle-interferometry we have carried out repeated measurements of relative positions for the components of 34 T Tauri binary systems. The projected separation of these components is low enough that orbital motion is expected to be observable within a few years. In most cases orbital motion has indeed been detected. The observational data is discussed in a manner similar to Ghez et al. (1995AJ....110..753G). However, we extend their study to a larger number of objects and a much longer timespan. The database presented in this paper is valuable for future visual orbit determinations. It will yield empirical masses for T Tauri stars that now are only poorly known. The available data is however not sufficient to do this at the present time. Instead, we use short series of orbital data and statistical distributions of orbital parameters to derive an average system mass that is independent of theoretical assumptions about the physics of PMS stars. For our sample this mass is and thus in the order of magnitude one expects for the mass sum of two T Tauri stars. It is also comparable to mass estimates obtained for the same systems using theoretical PMS evolutionary models.