Combine sake, soy sauce, rice vinegar, sugar, ginger and garlic in medium-size nonreactive saucepan. Bring to boiling over medium heat. Add mushrooms. Reduce heat to medium-low; simmer until liquid is almost evaporated, 25 to 30 minutes. Remove from heat.
Remove and discard ginger. Stir in sesame oil. Set aside to cool to room temperature. Cover, refrigerate and serve cold. Makes 4 side-dish servings.