Summary: We investigate the one-dimensional symmetric Schrödinger operator \(H_{X,\beta}\) with \(\delta^{\prime}\)-interactions of strength \(\beta = \{\beta_n \}_{n = 1}^{\infty } \subset \mathbb R\) on a discrete set \(X = \{ x_n \}_{n = 1}^{\infty } \subset [0, b)\), \(b \leq + \infty\) (\(x_n \uparrow b\)). We consider \(H_{X, \beta}\) as an extension of the minimal operator \(H_{\min}:= - d^{2}/dx ^{2}\lceil W_{0}^{2.2}(\mathbb R \setminus X)\) and study its spectral properties in the framework of the extension theory by using the technique of boundary triplets and the corresponding Weyl functions. The construction of a boundary triplet for \(H _{\min}^{*}\) is given in the case \(d_{*} := \inf_{n \in \mathbb N} |x_n - x_{n-1} | = 0\). We show that spectral properties like selfadjointness, lower semiboundedness, nonnegativity, and discreteness of the spectrum of the operator \(H_{X, \beta}\) correlate with the corresponding properties of a certain Jacobi matrix. In the case \(\beta_n > 0\), \(n \in \mathbb N\), these matrices form a subclass of Jacobi matrices generated by the Krein-Stieltjes strings. The connection discovered enables us to obtain simple conditions for the operator \(H_{X, \beta}\) to be selfadjoint, lower semibounded and discrete. These conditions depend significantly not only on \(\beta \) but also on \(X\). Moreover, as distinct from the case \(d_{*} > 0\), the spectral properties of Hamiltonians with \(\delta\)- and \(\delta^{\prime}\)-interactions in the case \(d_{*} = 0\) substantially differ.