Blowing it out very very well really seems to help me. And I mean blowing it out of the thread end (putting your mouth over the cart end). Some folks do it the other way around, but this really works best for me and gets no juice in your mouth.
If that doesn't work (and eventually it only does so much good) my deep cleaning regime is to put it in a small jar of white vinegar for a few hours (like 3-4hrs). Then about once an hour, really stir it up good. You can see the leftover juice dissolving into the vinegar when you hold up the jar to a light. It looks like heat waves or convection currents coming out of the atty. Afterwards, rinse it very well in warm/hot water, blow out as much water as you can, stick a paper towel up into it and leave it upside down overnight.
The next morning take a look at the little thread wicks coming from the bridge. If you did a really good job, they should be pure white like they were when the atty was brand new.
Hope it helps.