I don't rinse them. I put them all in a ziploc and throw them in the fridge as is. I'm at home but I do use the same set all day because I"m too lazy to wash them multiple times during the day. The only thing I change is the bottle so I'm not pumping warm milk into cold milk.
I keep a clean wash cloth in my pump bag and just wipe down the parts before I put them back in the bag. I find that leaving drops of milk on them makes them kind of greasy for the next pumping session.
I rinse and then either air dry or toss into a ziploc bag. I will however, pump into refrigerated milk if I only pumped a little the first time. I need to buy more medela bottles.
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Re: Those of you who refrigerate parts...
I do the same as PP...Just put in ziplock bags and put them in the cooler in the fridge at work, no rinsing or anything.
I use the same set all day, then wash them when I get home at night.
TTC since 11/05...ectopic pg 4/08...early m/c 6/09...BFP 10/5/09!

Nora B...June 15, 2010...8lbs, 8oz...Med-free birth!
TTC #2 since 7/11...cycle #3 of Clomid + IUI = BFP

Malcolm...September 21, 2012...8lbs, 6oz...Another med-free birth!
I rinse them twice a day (as in every 12 hours) and just toss them in the fridge otherwise.
I only have one set, except for the bottles.