Hey maddog. I know exactly what you mean. One thing that helps me is to mute the string right before moving to the next note. You can do this with one or both of your hands. If you let up on the string that your are fretting slightly with your fretting hand just enough to not be pushing it down it will be muted. Keeping your fingers close to the fretboard can make this easier for you. Sometimes I also use my picking hand to mute the strings too. The three fingers that I don't hold the pick with and the palm of my hand really help to shut the strings up when they should be quiet. Hope I helped a bit.

Laters,
Nate