I don't have any fancy traditional teapots, so I mostly clean in the dishwasher. I use a Teavana's "Perfect Teamaker" 32 oz. and also a plain clear glass teapot, both of which are dishwasher-safe. I rinse them after each use, and I run them through the dishwasher about once a week.
One interesting thing I learned about the Perfect Teamaker which might be of use to anyone using one of this kind with the built-in filter... one of the issues people sometimes have is difficulty in cleaning the filter thoroughly in order to eliminate flavors/odors between types of teas. I, being a newbie dimwit, unthinkingly put Lapsang Souchong in mine.... NOT a good idea. I ran it through the dishwasher... no good. I soaked it overnight with hot water and a bit of bleach... still smelled smoky. Then I tried baking soda and hot water, and Presto! It worked.
So now, I clean it this way after any strongly flavored/fragranced tea.
But no, I'm not a fanatic.
Drinking my 3rd infusion of TKY from last night's leaves. Still yummy.