It may help to use a mesh bag or strainer. This doesn't precisely prevent the sludge, but it can help one avoid it in their drink.
mesh bags, or even tea bags, are available and often used for such purposes - allowing flavor to infuse out while keeping the less-desirable parts (like the fiber in cinnamon) in the bag, easy to pull out and dispose of.
It may be wise to consider pre-preparing your cinnamon infusion or tisane, especially if making a complex or thickly textured drink - this will let you get the cinnamon flavor without problems like heating ingredients not meant to be heated in a cold drink, or letting flavor infuse more easily into a thin liquid rather than a thick one.