Great idea, I will do this the next time. I bet a coffee filter would work well too.
Best thing I've ever bought for cleaning my chain are those value packs of microfiber cleaning cloths you can get at auto stores or home depot. They pick up dirt and grease soooo much better than t-shirts. I also will use old tooth brushes and I do have a chain/cassette cleaning "wand" with scrubbies to loosen any really dried bits of grime.
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I'm a member of the "rag and degreaser" club, since I have an endless supply of old t-shirts. I agree with Reesha about the microfiber cloths though- they work great!
Many years ago, I tried a chain cleaner. I found it to be messy and one more thing to clean after use. YMMV.