Here are two places I tried and tested.
C'era una volta una Piada
I really just happened upon this place, which looked like someone's grandmother's kitchen. I was actually looking for a lunch cafe I found on a blog and couldn't find it, when I saw this cafe.
They serve these flatbreads which are typical from the Romagna region and you can fill it with whatever you like. The dough is kind of like a pizza dough, but it is thinner.
I went for one filled with grilled aubergines, courgettes, mushrooms and cheese and it more than filled me up too.
Another place I found was slightly more of this time in terms of its interior. Modern and minimalist with a communal table in the middle. They serve breakfast as well as lunch and offer sandwiches salads and delicious looking pastries.
I went for a grilled ham and cheese sandwich, Italian style of course; prosciutto and fontal cheese on a thick slice of sourdough bread, smeared with home made mayo. Loved it.