So I've been trying to explore more of the Japanese teas out there, however I keep running into a recurring problem--namely, the green tea-sludge left at the bottom of my pot when I'm done. I don't know, maybe it's just me, but I find it very obnoxious (and more than a little gross) to clean out--it doesn't come out neatly like whole leaves do, and since I know better than to dump the stuff in my garbage disposal, I end up having to scoop out what I can of the green sludgy mess out of the pot with my fingers before rinsing out the rest, which is not exactly a fun process.
So maybe this is a bit of a silly question, but--am I doing this completely wrong? Is there any easier/better way to do this? Some tool or technique I should use? I do I just need to resign myself to green sludge being the price I pay for a tea-drinking session?