What are some interesting (i. e. inexpensive) ways of getting from Paris to Rome besides flying?

The train is probably the other 'interesting' alternative. It is takes at least 12 hours. It may be less expensive although it may be less to take a low cost airline. You can check the low cost airfares using sites such as Kayak.

The fastest train will go through Zurich-Milan-Rome. You can also do this as an overnight. To quickly check the schedules, the German Rail site is useful but it will not give you prices. For prices, use the French train site. Kevin has instructions on how to use the site and trains in France on Slow Travel.

Bonne Chance.

I'll second Marta's suggestion. The buses take too long and are not very comfortable. The train is am interesting alternative but you either "lose" one day traveling to see the scenery or do it overnight where you might have to do a transfer or two. Cost-wise I don't think it makes a lot of difference from a low budget airline like easyjet where depending how far ahead you book, tickets could be as low as 40 euros.

A few years ago, we took the overnight from Paris to Milan; spent 4 days in Varenna; took the train from Milan to Rome (day). It was fine...there were 4 of us. Our sons had one compartment and we had another. Slept well. Woke up early to see the Alps.

Cameron