Does anyone know of any search tools to find cheap places to stay in Italy besides Airbnb ?
There will be locally owned 2 star 'hotels'....the ones that used to have the bathrooms down the hall. Generally called pensione. Booking dot com often lists them. Booking allows you to sort by your price point and lists a variety of lodging options. Convent Stays is a website for just that - accommodations provided by religious orders. However, these are not as inexpensive as they used to be. Others have luck contacting the city tourist office as some people cannot afford the costs for listing on the web so the office might just have a list with emails.
My impression is that Airbnb is one of the least desirable ways to book due to higher prices and non-accommodating cancellation policies.
We often use www.Booking.com as one can filter by price and size, among other factors. As Maria mentioned, a city’s tourism — especially for smaller towns - can be very fruitful for B&Bs that do not advertise through booking engines. Or try the lower-priced options in Rock Steves’ guidebook if he covers the place you want to go.
Where in Italy are you visiting? What is your budget?
Many BNB's and smaller often family run hotels are also listing on Airbnb. For example my next Italy trip I used Airbnb to book a bed and breakfast that was also listed on booking.com and Expedia as well. I have observed this is becoming more and more common.
try monasterystays.com