In a medium bowl, combine grated carrots and brown sugar. Set aside for 60 minutes. Drain.
Hydrate the raisins by letting them sit in a cup with the drained pineapple juice and rum. Drain.
Preheat oven to 350 degrees F (175 degrees C). Grease and flour two 10 inch cake pans, or three 8" pans.
In a large bowl, beat eggs until light. Gradually beat in the white sugar, oil and vanilla. Stir in the pineapple. Combine the flour, baking soda, salt and cinnamon, stir into the wet mixture until absorbed. Finally stir in the carrot mixture and the walnuts. Pour evenly into the prepared pans.
Bake for 45 to 50 minutes in the preheated oven, until cake tests done with a toothpick. Cool for 10 minutes before removing from pan. When completely cooled, frost with cream cheese frosting.