Does cloth retain meat and milk after it is washed?
I was thinking that instead of ceramic or plastic plates, we could eat off of woven disks.
Sefer Q&A by Rabbi Wiesenfeld (page 68) quotes Rabbi Ovadia that washing (or maybe even soap itself) would remove the meat or milk status of a cloth.
Rabbi wiesenfelds points out, that the halacha of not intentionally converting milky utensils to meat and vice versa should also apply here. This in the addition to the custom he mentions, that says that we should have different tablecloths for milk and meat would render your suggested solution a not good one.
Sefer הכשרות page 38 also says that the minhag is to have separate tablecloth. Halchically speaking he agrees that washing is enough. He bases this on the rema in hilchos pessach 448:11 who allows washing of rags that held flour.