Am I an optimist?

A lot depends on what time of year you will be traveling. Also, sounds like you're looking for apartments? Or do you want B&Bs?

Put the approximate dates you wish to travel on into a site such as booking.com
This will give you a good feel for what you can get and for how much.

You need to explain "avoid hotels." What do you want? A lot will depend on location. $150 could be optimistic in Rome.

You might want to look into guest houses run by monasteries and convents (check monasterystays.com).

Look on VRBO and Airbnb. I found nice 2 bedroom apartments in all three locations for ~$175. We haven't gone yet, so I can't recommend our particular accommodations, but I was surprised at how many I found in that price range when I was looking.

With the current favorable exchange rate you can easily fit within this budget. I target Euro 100 per night for our travels, occasionally going over and many times under. Apartments are a good bet for 7 night stays (actually anything over 3 nights) and you further save money by eating in a bit. Booking.com and VRBO.com are my favorites, but sometimes I just Google "B&B in XXXX" or "apartment in YYYY" and find gems not listed on other sites. I recently did that for Amsterdam and scored a terrific B&B not listed elsewhere. (The smaller places really eschew booking sites because they have to pay a fee.)

My husband and I traveled to Italy with a budget of 100 euro per night in 2010. It was quite manageable. With the favorable exchange rate this year staying within a budget of 150 USD or less should be easy!

Are you interested in apartments instead of hotels? Or B&B's over hotels?

If you don't mind a small family run hotel. We loved Hotel Italia Roma. (120 euro per night for a double) it was clean, quiet, comfortable and within easy walking distance of the busses and metro which will take you to all the sights. http://www.hotelitaliaroma.it/en/ Gabriel at the front desk helped us reserve tickets to the Borghese Gallery, let us know which sites were open/closed that particular day and gave detailed maps and instructions for bus routes. We felt like part of the family!

I like to use www.bedandbreakfast.it
as an alternative to hotels. You can search around points of interest and a wide range of prices is available. There's a free App for that website.

Not sure if you'd like convents or monasteries. Mattresses are hard. Rooms are sparse, No TV and No WiFi in rooms. You have to go to their lobby for these services. Breakfasts can be plain, weak coffee and rosetta buns, every morning. Ruined me for Rosetta buns forever. I would sneak out at 6 or 7 AM and go to the Bar across Via Merulana in Roma for a caffe' and pastry. I'm sure I was Not their model guest.

My experience with staying at religious organisations in Italy is different than that of Sandra. We enjoyed our simple yet tasty and filling breakfasts, coffee came out of an automatic coffee machine like in so many hotels and we could choose espresso, cappuccino or others, the juice was fresh, the ladies very helpful and the bathrooms compact but fully equipped (and the shower larger than the one we had at the Smeraldo, a hotel in Rome.

Yes, the TV was downstairs and I don't remember wifi, but this was before wifi when a cable was required for ethernet and the place to get internet was an internet cafe and you had to turn over your first born for a deposit.

The beds were comfortable, and there was no trouble pushing them together. In fact the cleaning Sisters left them that way while we were out.

Def look into airbnb...
We stayed at this lovely apartment in Rome last Sept with views from the window of the Colosseum and just steps away...
https://www.airbnb.ca/rooms/2193487

Three of us, with the airbnb fee, it was about $725 for 5 nights, and he gave a 5% discount for booking 5 nights. And that was Canadian dollars - your American will go much further...but he is already booked up almost full to Sept!

In Venice, we stayed in this quiet little apartment...about a 5 min walk to Rialto, grocery and bank and restaurants in the campo...
https://www.airbnb.ca/rooms/1683211

With fee, $465 canadian for 3 nights

We've also used this B&B for our other three previous visits...
http://alcampaniel.tripod.com/

Standard double was about 70-75 euro a night (in Sept).

In Florence, we stayed at this place - a little more off the beaten path, but the bus gets you most of the way there...
https://www.airbnb.ca/rooms/677883
We actually had an amazing rate for an apartment - $100 total - with fees! - for 2 nights, but her prices have gone up since 2012, but still a great deal.

Depending on the time of year, some places could already get booked up - especially if you want a whole week - could be hard to find - best of luck - there are gems out there, it just takes time and effort to find them :)

I think the suggestions of Bookings.com is a good idea. I just booked two weeks in May in the popular cities and the prices have gone up since our last trip in 2008. I was really surprised since Europe always ran cheaper than the US. But prices here have gone up too. With any luck the Euro will stay low and you can make it up that way.

You might consider looking at hostels, too. Some of them have private rooms with private baths. An example is in Florence, look into the Ostello Archi Rossi. We stayed in the Annex. Had a double room with bath. There was also a wonderful rooftop terrace across from our room. While it was not private, it was rarely used by other guests. Caution - there are A LOT of stairs.

In Rome, try The Beehive. They have private rooms, too. It looks like, there, you might need to share a bathroom.

Enjoy,
Sharon

I would like to second Nigel's experience with convents. I have had universally good experiences with them. There are some historically interesting buildings in prime locations, generally for around €50 a night for a single. They are always spotlessly clean. Breakfasts vary, as they do in budget hotels, but the best food I've ever had in Italy was in a monastery in Le Marche. And the sisters have been very kind to me.

You mentioned in another message needing to avoid stairs – most have elevators because they have some elderly members in the community. But don't expect a TV in your room or lots of amenities, and you often won't find much English spoken. But that's never had a negative impact on my stays