Granular secondary particles of a lithium-manganese composite oxide suitable for use in non-aqueous electrolyte secondary batteries showing high-output characteristics which are granular secondary particles made up of aggregated crystalline primary particles of a lithium-manganese composite oxide and have many micrometer-size open voids therein with a defined average diameter and total volume of open voids. A process for producing the granular secondary particles which includes spray-drying a slurry of at least a manganese oxide, a lithium source, and an agent for open-void formation to thereby granulate the slurry and then calcining the granules.