  Molecular line observations using a variety of tracers are often used to
investigate the kinematic structure of molecular clouds. However, measurements
of cloud velocity dispersions with different lines, even in the same region,
often yield inconsistent results. The reasons for this disagreement are not
entirely clear since molecular line observations are subject to a number of
biases. In this paper, we untangle and investigate various factors that drive
linewidth measurement biases by constructing synthetic
position-position-velocity cubes for a variety of tracers from a suite of
self-gravitating magnetohydrodynamic simulations of molecular clouds. We
compare linewidths derived from synthetic observations of these data cubes to
the true values in the simulations. We find that differences in linewidth as
measured by different tracers are driven by a combination of density-dependent
excitation, whereby tracers that are sensitive to higher densities sample
smaller regions with smaller velocity dispersions, opacity broadening,
especially for highly optically thick tracers such as CO, and finite resolution
and sensitivity, which suppress the wings of emission lines. We find that, at
fixed signal to noise ratio, three commonly-used tracers, the J=4->3 line of
CO, the J=1->0 line of C^{18}O and the (1,1) inversion transition of NH_3,
generally offer the best compromise between these competing biases, and produce
estimates of the velocity dispersion that reflect the true kinematics of a
molecular cloud to an accuracy of about 10% regardless of the cloud magnetic
field strengths, evolutionary state, or orientations of the line of sight
relative to the magnetic field.
