For a finite-dimensional Lie algebra L over C with a fixed Levi decomposition L = g proportional to tau , where g is semisimple, we investigate L-modules which decompose, as g-modules, into a direct sum of simple finite-dimensional g-modules with finite multiplicities. We call such modules g-Harish-Chandra modules. We give a complete classification of simple g-Harish-Chandra modules for the Takiff Lie algebra associated to g = sl(2), and for the Schrodinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright's and Arkhipov's completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple g-Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff sl(2) and the Schrodinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple g-Harish-Chandra modules.