  We calculate in (two-loop) chiral perturbation theory the local NN-potentials
generated by the three-pion exchange diagrams with one insertion from the
second order chiral effective pion-nucleon Lagrangian proportional to the
low-energy constants $c_{1,2,3,4}$. The resulting isoscalar central potential
vanishes identically. In most cases these $3\pi$-exchange potentials are larger
than the ones generated by the diagrams involving only leading order vertices
due to the large values of $c_{3,4}$ (which mainly represent virtual
$\Delta$-excitation). A similar feature has been observed for the chiral
$2\pi$-exchange. We also give suitable (double-integral) representations for
the spin-spin and tensor potentials generated by the leading-order diagrams
proportional to $g_A^6$ involving four nucleon propagators. In these cases the
Cutkosky rule cannot be used to calculate the spectral-functions in the
infinite nucleon mass limit since the corresponding mass-spectra start with a
non-vanishing value at the $3\pi$-threshold. Altogether, one finds that chiral
$3\pi$-exchange leads to small corrections in the region $r\geq 1.4$ fm where
$1\pi$- and chiral $2\pi$-exchange alone provide a very good strong NN-force as
shown in a recent analysis of the low-energy pp-scattering data-base.
