  In the framework of the seesaw mechanism with three heavy right-handed
Majorana neutrinos and no Higgs triplets we carry out a systematic study of the
structure of the right-handed neutrino sector. Using the current low-energy
neutrino data as an input and assuming hierarchical Dirac-type neutrino masses
$m_{Di}$, we calculate the masses $M_i$ and the mixing of the heavy neutrinos.
We confront the inferred properties of these neutrinos with the constraints
coming from the requirement of a successful baryogenesis via leptogenesis. In
the generic case the masses of the right-handed neutrinos are highly
hierarchical: $M_i \propto m_{Di}^2$; the lightest mass is $M_1 \approx 10^3 -
10^6$ GeV and the generated baryon-to-photon ratio $\eta_B\lesssim 10^{-14}$ is
much smaller than the observed value. We find the special cases which
correspond to the level crossing points, with maximal mixing between two
quasi-degenerate right-handed neutrinos. Two level crossing conditions are
obtained: ${m}_{ee}\approx 0$ (1-2 crossing) and $d_{12}\approx 0$ (2-3
crossing), where ${m}_{ee}$ and $d_{12}$ are respectively the 11-entry and the
12-subdeterminant of the light neutrino mass matrix in the basis where the
neutrino Yukawa couplings are diagonal. We show that sufficient lepton
asymmetry can be produced only in the 1-2 crossing where $M_1 \approx M_2
\approx 10^{8}$ GeV, $M_3 \approx 10^{14}$ GeV and $(M_2 - M_1)/ M_2 \lesssim
10^{-5}$.
