This sounds like pseudoscience, but it appears in Journal of Neuroscience, so … Weirdly, a rat study has found that sleeping on the side (the most common posture for humans and other animals) is the best position for efficiently removing waste from the brain.
Brain waste includes amyloid-beta and tau proteins, whose build-up is a critical factor in the development of Alzheimer's disease.
The study used imaging of the glymphatic pathway, which clears waste products from the brain by filtering cerebrospinal fluid through the brain and exchanging it with interstitial fluid. The process is most efficient during sleep, and its efficiency is affected by the level of consciousness. The researchers compared glymphatic transport during sleep when anesthetized rodents’ brains were in three positions—lateral (side), prone (down), and supine (up).
Of course, these findings need to be confirmed in humans (which might be tricky!), but there is, after all, no harm in changing your sleep position, if you don't already sleep on your side (though I concede it can be a difficult thing to change).
Apart from providing a practical tip for fighting age-related cognitive decline and dementia, the finding also supports the idea that one of the purposes of sleep is to ‘clean up’ the mess that accumulates while we are awake.
The finding is also consistent with increasing evidence that sleep disturbances are a factor in the development and progression of dementia.
 . The Effect of Body Posture on Brain Glymphatic Transport. The Journal of Neuroscience [Internet]. 2015 ;35(31):11034 - 11044. Available from: http://www.jneurosci.org/content/35/31/11034