If you haven't already done so use iTunes > Library > Organize Library > Rearrange files in the folder 'iTunes Music'. This moves any iPod games, mobile apps and ringtones into the main media folder, and moves music into it's own subfolder like other media types. If the option is greyed out it has already been done.
Once done you can close iTunes, rename iTunes Music as iTunes Media then reopen. This will make your library match the current layout. Check that media folder preference updated automatically. If not, change it, then close and reopen iTunes.
With iTunes closed you can now move or copy the entire iTunes folder to your E: drive as E:\iTunes. When the transfer is complete press and hold down the shift key as you click the icon to launch iTunes. Keep holding until asked to choose or create a library. Click choose and browse to the file E:\iTunes\iTunes Library.itl.
If you don't use the rearrange library bit above and your library is in the old layout (iTunes Music is a hint that it might be) then iTunes will still look for iPod Games, mobile apps and ringtones (now just tones) on the C: drive.
See make a split library portable for a more generic version of the instructions above.
See also this backup tip. Might be best to copy the current library after getting it in the right shape and not delete the original until you've made a backup on a separate drive.
tt2
Message was edited by: turingtest2