Electrochemical cells that use electrolytes made from new polymer compositions based on poly(2,6-dimethyl-1,4-phenylene oxide) and other high-softening-temperature polymers are disclosed. These materials have a microphase domain structure that has an ionically-conductive phase and a phase with good mechanical strength and a high softening temperature. In one arrangement, the structural block has a softening temperature of about 210° C. These materials can be made with either homopolymers or with block copolymers. Such electrochemical cells can operate safely at higher temperatures than have been possible before, especially in lithium cells. The ionic conductivity of the electrolytes increases with increasing temperature.