An LCK manifold is a Hermitian manifold \((M,g,J)\) such that the fundamental \(2\)-form \(\Omega\), defined by \(\Omega(X,Y)=g(X,JY)\), satisfies the condition \(d\Omega= \omega\wedge \Omega\) for a closed 1-form \(\omega\). An LCK manifold is said to be Vaisman if \(\omega\) is parallel. In this paper the author gives necessary and sufficient conditions for a LCK structure on a strongly unimodular Lie algebra to be Vaisman. Let \(G\) be a simply-connected solvable Lie group and \(\Gamma\) a discrete co-compact subgroup. The compact quotient manifold \(\Gamma\backslash G\) is called a solvmanifold. The above-mentioned conditions can be applied to any solvmanifold with a left-invariant LCK structure. As a corollary, it follows that Inoue surfaces and Oeljeklaus-Toma manifold have no Vaisman structures such that the complex structure is left-invariant.