Rome and Amalfi Coast
I'm not sure where on the Amalfi coast you are going, but if you plan to stop in Positano, my wife and I really enjoyed Barilotto del Nonno(translates to my grandfather's barrel). It is a family owned tratorria style restaurant and the food was fantastic and very authentic. The service was warm and friendly as well. It also definitely is a non-touristy location. When we went we were one of two parties at the restaurant. It's a bit off the beaten path, but worth going to.
i'm sure there are always new suggestions and hopefully they will be posted but in the meantime - there are literally hundreds and hundreds of postings on both rome and the amalfi coast. i recently read thru many of them for a friend's trip. it's actually kind of fun (tho a bit time consuming) because you get to read so many varying opinions on some of the same places and also get to read such a widespread survey.
fyi - one of my most favorite places in rome is la campana right off piazza navona. order what looks good as it's being served to other tables. works every time. and on the amalfi coast - trattoria armandino in marina del praia right before you get to positano - it's this teeny little fishing village with a wonderful restaurant. it's the last one on the right before you get to the little wharf & cliff walkway. not at all fancy. their spaghetti alle frutti de mare & home cured anchovies are great. very local.....!