  We study the spatio-temporal behavior of the Els\"asser variables describing
magnetic and velocity field fluctuations, using direct numerical simulations of
three-dimensional magnetohydrodynamic turbulence. We consider cases with
relatively small, intermediate, and large values of a mean background magnetic
field, and with null, small, and high cross-helicity (correlations between the
velocity and the magnetic field). Wavenumber-dependent time correlation
functions are computed for the different simulations. From these correlation
functions, the decorrelation time is computed and compared with different
theoretical characteristic times: the local non-linear time, the
random-sweeping time, and the Alfv\'enic time. It is found that decorrelation
times are dominated by sweeping effects for low values of the mean magnetic
field and for low values of the cross-helicity, while for large values of the
background field or of the cross-helicity and for wave vectors sufficiently
aligned with the guide field, decorrelation times are controlled by Alfv\'enic
effects. Finally, we observe counter-propagation of Alfv\'enic fluctuations due
to reflections produced by inhomogeneities in the total magnetic field. This
effect becomes more prominent in flows with large cross-helicity, strongly
modifying the propagation of waves in turbulent magnetohydrodynamic flows.
