Please share with us any less well known museums, restaurants, ancient sites, or anything else you may have enjoyed in Rome. We will have the good fortune to stay in Rome for five weeks, in an apartment, in September - October. We have visited Rome before and been to all the iconic attractions. We also have Rick's 2013 book, so there is no need to suggest anything therein. Thank you!
Here's an excellent book that's exactly what you're looking for: http://www.amazon.com/City-Secrets-Rome-Robert-Kahn/dp/1892145049 And there's this blog: http://romethesecondtime.blogspot.com/ We're coming up on our fourth trip to Rome so, like you, we've been looking for the less famous places. Check out Villa Torlonia, as an example.
We stayed in a bed and breakfast in Zagarolo, about 20 miles south of Rome's city center. Our host has us convinced that there is as much to see in the suburbs as there is in the city. We were carried to the top of a 2500' mountain, and hiked down ancient walking paths to a pagan temple of Palestrina. The temple was uncovered by allied bombs in WW II, after being built over. The town of Zagarolo was where Roman solders' helmets were made. It was also where gladiators had gymnasiums for training.
There are all kinds of gardens in the suburbs, some of which were Summer places for the popes. Rural Rome is just full of treasures to see.
On a Saturday morning, the Palazzo (Galeria) Colonna and, especially, the Princess Apartments therein. Not cheap. Worth every euro. And then some. I have no idea if it is in the book so if it is just ignore.
Some of the less well-known churches might be interesting: Quattro Coronati near San Clemente, Santa Prassede near Santa Maria Maggiore. The church complex os Santa Costanza on Via Nomentana- beautiful mosaics in the ruins of what was a large complex. Walk part of the wall near the San Sebastiano gate. Visit the Jewish Ghetto and have a kosher King David pizza (pizza Re David), visit the Synagogue and museum (Rome has the oldest documented Jewish community in Europe). Then just wander.
As mentioned, Villa Torlonia and Palazzo & Galleria Colonna, but also Villa Medici, Palazzo Farnese, Palazzo Braschi. The latter 3 are not far off the beaten path, but less visited and under-appreciated. Also Parco degli Acquedotti is a nice outing with magnificent vistas. You might look for exhibitions at Chiostro Bramante and Palazzo della Esposizioni as well as other galleries. With a nice long stay you can take time to enjoy some of the temporary exhibits at various galleries and locations.
Another excellent book I'm enjoying is Frommer's 24 Great Walks in Rome:
http://www.amazon.com/Frommers-24-Great-Walks-Rome/dp/0470928166 It contains a lot of unusual sights to see.
I am glad Michael mentioned "24 Great Walks in Rome." It guided us for two trips here prior to relocating, and we still use it occasionally.
We liked visiting the Testaccio neighborhood (though it is in Rick's book we didn't see many tourist-types there) - the Protestant Cemetery is fun to wander around on a nice day, the pyramid nearby is amazing, and there is an interesting market. We returned one evening for dinner at Perilli's Restaurant and had a delicious meal!
I agree with the testaccio market and area (you can find excellent olive oil at the market). If you go, you might as well check out the key hole view of St Peter's church on Avetino hill, pretty cool. If you want a change from Italian food, go to Cantina Luciferno in Campo dei Fiori, great food and service.
Ex-pats who are living in a new country are discovering and writing about the delights of Italy every day. Luckily, this Italian Reflections site provides an index to them: http://www.italianreflections.com/english-language-blogs-from-italy