Bring 1 cup of water to a boil. Add rice and bring to a boil again. Once boiling, cover, reduce heat to low, and simmer until rice is cooked, about 40 minutes. The rice will be soft and waterlogged; drain off any excess water if necessary.
Transfer rice to a blender and add 2 cups of warm water. Blend until well incorporated, about 2 to 3 minutes. Add another 1 cup of warm water and blend again. If you prefer an even thinner non-dairy milk, add another 1/2 to 1 cup of warm water and blend again.
Let the mixture sit for 30 minutes. Meanwhile, drape cheesecloth over the top of a pitcher or storage container and secure it around with a rubber band. Make sure the cheesecloth drapes down into the container and is not tightly fashioned straight across, as it needs to act as a strainer.
After 30 minutes, use a spoon to scoop any residue off the top of your milk mixture (usually a thin film forms during the cooling process). Once that is removed, slowly pour the milk into the container through the cheesecloth. Be sure not to pour all the bits that have collected at the bottom.
Remove rubber band and, while holding the cheesecloth closed, gently squeeze out any last liquid. Add 1 tsp of vanilla extract and 2 tsp of sweetener such as agave, pure maple syrup, or brown rice syrup. Taste adding more sweetener as desired. Chill for several hours before drinking.
Chef's Note: You can buy a 100% natural cheesecloth at cooking stores or health food stores. The tighter woven it is, the better (mine is ultra fine). You need a size that is wide enough to cover your container two times.