Must see's that aren't the regular tourist sites

You could throw a dart really. I typically search all of the possible flights and work up a trip around what I find. Ancona led me to Marche for example (fabulous white Verdicchio matelica). I'm a little weird in that I've been to Italy seven times but not to Rome, Florence, or Venice, but there is stuff to see in every region! At a quick glance, I see Piedmont, Liguria, Umbria missing from your list. Molise and Calabria are really the only place that raise eyebrows--you don't hear much about the very far north either.

We understand Tuscany needs a car.

The Tuscany region is a clichéd regular tourist area

The Tuscany region is a clichéd regular tourist area

Not necessarily so, check out the mountainous Garfagnana or the Etruscan ruin-studded coastal Maremma.

I've noticed that the stunning Roman mosaics at Ravenna do not get as much attention from Americans, the town itself is quite charming and merits a night or two on it's own.

i didn't see sicily on your list.

I hear Bologna in the Emilia Romagna region is amazing, especially with food but also has interesting sights too. Stanley Tucci’s recent food show highlighted Bologna and most of the 20 regions of Italy. Rick’s radio show on October 1, has a conversation with Fred Plotkin who has lived in Bologna and written extensively about the food. Sounds like an area to explore that’s mostly untouristed.
You might check it out.

We don’t go for “sites;” We go for experiences. Have you considered basing yourself in a town for a week and just enjoying life there? Bologna would be a good base to enjoy and offers many options for day tripping to see other towns like Ferrara, Ravenna, etc., without a vehicle. We have spent a week in Camogli in Liguria and really enjoyed small town life there with the option to visit some stunning villages and vistas.

JR

If you haven't been to Ravenna, please consider visiting. There is no better place to marvel at the development of Byzantine art and architecture (in a self-contained, walkable city). San Vitale, the so-called Mausoleum of Galla Placidia, San Apollinare Nuovo, San Apollinare in Classe, the two Baptisteries and more. If you like mosaics, this is well worth your time. I have not stayed there but traveled by train from Bologna.

I've stayed in Lucca twice for extended stays, and haven't seen that many tourists, and never saw a tour group once.
This was pre pandemic too.
It's just a lovely town with plenty to see and do; and lots nearby, accessible by bus or train.

As others said, there are several hidden gems for each italian region, just to mention some of them near my home town in veneto: the cities of Padua, Treviso, Vicenza, the hill towns of Bassano del grappa and Asolo, the practically unknown (to tourists) region of Friuli-Venezia- Giulia with its major historical cities of Udine, Pordenone, Cividale del Friuli, Trieste etc.
Places off the beaten path can be quite challenging to explore but at least you can feel the essence of the real Italy and all its cultural diversities from region to region, and not the usual cliches that seem to reduce italy to the only land of pizza, pasta and red & white tablecloths

The island of Procida is beautiful. It is not as touristy as Capri or Ischia. You can take a ferry from the port of Pozzuoli which is closer then leaving from Naples. You can also check out the Roman Flavian Amphitheater in Pozzuoli. I would also highly recommend seeing the ruins of Cuma. They are also a short drive from Pozzuoli and contain the cave of the Sibyl, which was one of the most famous oracles in the ancient world. You would probably have the ruins to yourself.

First of all is important remember that there are several areas in Italy that receive a lot of tourists but aren't the usual touristic places for travellers from US. Garda's Lake for example is plenty of German and Austrian tourists, while Americans go to Como's lake primary.
Said that, the problem is that the whole Italy deserve to be visited and you'll find different dishes, food, ingredients, astonishing castles, historical cities and towns everywhere in Italy.
I'm from Emilia, so I can say that my Region could be a stop for you, but why not Apulia, Sicily, Marche, Veneto, Piedmont, Friuli, Trentino, Sardinia.... and I can continue with the other Regions.

Thank you all. This is exactly what we were looking for. Great suggestions and we are busy looking them all up. Ravenna seems to get several shout-outs and had never considered it. The fun for us this trip is to explore the off-the-beaten path. After a review of all the above suggestions, we are sure to have more questions! Thanks everyone!

Ravenna would pair nicely with Emilia-Romagna, with the Veneto (Chioggia is a place that always interested me), or northern Marche.