Since I clean and repair stethoscopes on the side here is how I claen them and what I suggest to to clean them.
First remove all your parts, if you can (ear pieces, chast piece, name tag, etc.)
Do a basic wipe down with a little bit of alcohol, bleach wipe (will not harm Littmans), equipment cleaning wipes.
If you are able to remove the ear and chest pieces you can do a soap and water flush through the inners of the stethoscope and let it air dry for a few days during your days off. Hang the stethoscpe on a hanger and let gravity do it's thing.
Take apart the chest piece and clean the rim, chest plates, and bell with the alcohol, bleach wipe, or equipment wipes.
As for the ear peices you can take the wipes and roll them up and push them through the ear pieces. Depending on the ear pieces it is just better to let them saok in soap and water and then wipe them with some type of wipe. In most cases it is better to get replacement ear pieces. They are cheap and you can toss them.
I hope this helps with cleaning instructions. If you have any other questions just let us know what they are.