  In the seesaw framework, in the basis of $M^{}_{\rm R}$ being diagonal, we
explore the simplified textures of $M^{}_{\rm D}$ that can naturally yield the
trimaximal neutrino mixings and their consequences for the neutrino parameters
and leptogenesis. We first formulate the generic textures of $M^{}_{\rm D}$
that can naturally yield the trimaximal mixings and then examine if their
parameters can be further reduced. Our analysis is restricted to the simple but
instructive scenario that there is only one phase parameter $\phi$ responsible
for both the CP violating effects at low energies and leptogenesis. Our
attention is paid to the textures of $M^{}_{\rm D}$ that possess some vanishing
or equal elements. On the basis of these results, we further examine if $\phi$
can also take a particular value. The consequences of the
phenomenologically-viable simplified textures for the neutrino parameters and
leptogenesis are studied. A concrete flavor-symmetry model that can realize one
representation of them is given.
