  The structure and star formation activity of a molecular cloud are
fundamentally linked to its internal turbulence. However, accurately measuring
the turbulent velocity dispersion is challenging due to projection effects and
observational limitations, such as telescope resolution, particularly for
clouds that include non-turbulent motions, such as large-scale rotation. Here
we develop a new method to recover the three-dimensional (3D) turbulent
velocity dispersion (${\sigma}_{v,3D}$) from position-position-velocity (PPV)
data. We simulate a rotating, turbulent, collapsing molecular cloud and compare
its intrinsic ${\sigma}_{v,3D}$ with three different measures of the velocity
dispersion accessible in PPV space: 1) the spatial mean of the 2nd-moment map,
${\sigma}_i$, 2) the standard deviation of the gradient/rotation-corrected
1st-moment map, ${\sigma}_{(c-grad)}$, and 3) a combination of 1) and 2),
called the 'gradient-corrected parent velocity dispersion',
${\sigma}_{(p-grad)}=({\sigma}^2_i+{\sigma}^2_{(c-grad)})^{1/2}$. We show that
the gradient correction is crucial in order to recover purely turbulent motions
of the cloud, independent of the orientation of the cloud with respect to the
line of sight (LOS). We find that with a suitable correction factor and
appropriate filters applied to the moment maps, all three statistics can be
used to recover ${\sigma}_{v,3D}$, with method 3 being the most robust and
reliable. We determine the correction factor as a function of the telescope
beam size for different levels of cloud rotation, and find that for a beam FWHM
$f$ and cloud radius $R$, the 3D turbulent velocity dispersion can best be
recovered from the gradient-corrected parent velocity dispersion via
${\sigma}_{v,3D}=[(-0.29\pm0.26)f/R+1.93\pm0.15]{\sigma}_{(p-grad)}$ for
$f/R<1$, independent of the level of cloud rotation or LOS orientation.
