  We have obtained high-dispersion spectra for 256 pre-main-sequence stars in
the Orion Nebula Cluster in order to measure their projected rotational
velocities and study the rotational evolution and physical properties of young,
low-mass stars. Half the stars were chosen because they had known photometric
periods and half were selected as a control sample of similar objects without
known periods. More than 90% of the spectra yielded vsin(i) measurements,
although one-third are upper limits. We find strong evidence confirming the
assumption that the periodic light variations of T Tauri stars are caused by
rotation of surface spots. We find no significant difference between the
vsin(i) distributions of the periodic and control samples, indicating that
there is no strong bias in studying the rotation of young stars using periodic
variables. Likewise, the classical and weak T Tauri stars exhibit vsin(i)
distributions that are statistically the same. For stars with known period and
vsin(i), the mean value of sin(i) is lower than expected for a random
distribution of rotation axes. This could be due to errors in the quantities
that contribute to the sin(i) calculation or to a real physical effect. We find
that mean sin(i) has the expected value if we increase the stars' Teff values
by 400-600 K. Finally, we have calculated minimum radii for stars with vsin(i)
and period, and average radii for stars grouped by location in the H-R diagram.
We find evidence at the 3-sigma level that the radii of stars on similar mass
tracks are decreasing as the stars move closer to the ZAMS. [abridged]
