Cook: Combine berries, sugar, juice, and zest in a medium saucepan. Set over medium heat and cook, uncovered, until berries begin to soften. Use a potato masher or fork to mash the berries a bit, then increase heat to medium/high and let simmer (10 min for cranberries, 20-30 min for strawberries and blueberries).
Cool: Remove from heat and allow to cool (mixture will thicken as it cools).
Store in an airtight container in the fridge for about 1 week.