no reservations

We did that in 2013 (our last trip 😢) in May - no problems. I would research the next place during our last night in the city before and send an email for reservation. Never a problem. We like to stay near the train station - I use google maps to see what is near and look them up.

It would be nice to have a place reserved for the first night when you land (wherever that might be).
Other than that, you can wing it as much as you like. Just be aware that you might find difficulty finding accommodations in particular places and periods if there are special events. Venice is always a good idea to book ahead of time, at least if you want to stay in the historical center for a reasonable price, otherwise you might be forced to stay in Mestre or Lido or even Padua.
Everyplace else will have availability somewhere even on the day of arrival. However nowadays it is essential to have a mobile device with internet connection, so that you can find vacancies quickly using a search engine (e.g. expedia, booking, etc.) which will give you the available options without having to make dozens of phone calls like in the old days before the internet.

I would book the first and last nights. You can use booking.com to find places to stay when you are ready to move on, unless you want to spend time looking for rooms when you could be enjoying a city.

I don't get this -- why would you want to spend your limited valuable vacation time searching for a room? And the odds of getting a charming room or a good deal are much lower done on the fly. IN times past this was standard and train stations had offices which provided booking services. But with the internet this sort of thing is obsolete. If I were going to do this I would at least have rooms booked in larger cities and be prepared to flop at some Ibis on the toll road if nothing better came along.

October can also be a busy month in Italy, so you could find a lot of places booked up but March should be a bit quieter. Although you should be able to find accommodations, the spontaneous approach comes with a few caveats.....

• you'll have to waste time during your trip looking for places to stay, either before you arrive in a city or when you arrive.
• you may have to take what you can get at the time, which may not be in the price range or comfort level you prefer, or may be located far from the city centre which means more time and money for transportation.

I prefer to book accommodations in advance, because that provides greater choice in the hotel, location and price range. Since I know where I'm staying when I arrive in a new city, I can check-in quickly, drop my luggage and start touring right away. I've seen people wandering about various cities in Europe hauling a couple of gargantuan steamer trunks and looking for a place to lay their head for the night. No thanks!

Another advantage to booking online (even same day) is that you can check out photos and reviews, which takes a lot less time than wandering around with luggage in tow.