We theoretically demonstrate that a type-II class of tilted Dirac cones can emerge in generalized two-dimensional anisotropic lattice arrangements. This is achieved by introducing a special set of graphynelike exchange bonds by means of which the complete spectrum of the underlying Weyl Hamiltonian can be realized. Our ab initio calculations demonstrate a unique class of eigensolutions corresponding to a type-II class of Dirac fermionic excitations. Based on our approach, one can systematically synthesize a wide range of strongly anisotropic band diagrams having tilted Dirac cones with variable location and orientation. Moreover, we show that asymmetric conical diffraction, as well as edge states, can arise in these configurations. Our results can provide a versatile platform to observe, for the first time, optical transport around type-II Dirac points in two-dimensional optical settings under linear, nonlinear, and non-Hermitian conditions.