Provided is a curable organopolysiloxane composition that has high thermal conductivity while suppressing a change in hardness of the cured material after heat aging, and that provides an organopolysiloxane cured material that can maintain flexibility and stress relaxation even when a thermally conductive member made of the cured materials is used at high temperature. The disclosure also relates to a thermally conductive member made of the composition and a heat dissipating structure using the member. The composition comprises: (A) an alkenyl group-containing organopolysiloxane, (B) an organohydrogen polysiloxane, (C) a thermally conductive filler in an amount of from 60 to 90% by volume, (D) at least one type selected from fatty acids, fatty acid esters, and fatty acid metal salts in an amount of from 0.05 to 2.00 parts by mass per 100 parts by mass of Component (C), (E) a hydrosilylation reaction catalyst, and (F) a heat resistance imparting agent.