Rinse the rice under cold water until the water runs clear.
In a blender, combine the rice and water. Blend until the rice is broken up but not completely smooth.
Let the mixture sit at room temperature for at least 3 hours or overnight.
Strain the rice mixture through a fine-mesh sieve into a bowl, discarding the rice solids.
Stir in the almond milk, coconut milk, sugar, vanilla extract, cinnamon, and salt until the sugar is dissolved.
Pour the mixture into an ice cream maker and churn according to the manufacturer's instructions.
Transfer the ice cream to a container and freeze for at least 1 hour before serving.