Preheat oven to 350 degrees and lightly grease a square baking dish.
In a large bowl, gently mix raspberries, peaches, flour, sugar, lemon juice, and cinnamon without crushing the berries.
Pour the fruit mixture into the prepared baking dish.
In a separate bowl, mix oats, brown sugar, flour, and butter until crumbly and moist.
Sprinkle the topping evenly over the fruit mixture.
Bake for 30 to 40 minutes until the filling is bubbling and the peaches are tender.
Let the cobbler rest for 10 to 15 minutes before serving warm with ice cream or whipped cream.