Using general non-skew symmetric classical r-matrices with a spectral parameter, we construct quantum spin chains that generalize famous Gaudin spin chains. We construct new classes of non-skew symmetric classical r-matrices associated with special guasigraded Lie algebras, with second order automorphisms and with quasi-classical limit of the Reflection Equation Algebras correspondingly. In the simplest case of Lie algebra sl(2) we calculate the spectrum and the eigenvalues of the corresponding Hamiltonians using the algebraic Bethe ansatz technique.