Is it possible, yes. Will it take some extra effort, maybe. And keep mind that every brand/property can have different policies.
Practically every hotel will accept cash as final payment so that is the smallest issue.
Where guests have trouble is with the reservation or deposit. If you are a walk in, the reservation is problem is eliminated so what you have to prepare for is any cash deposit requirement which could be anywhere from 0-200 Euros depending on the class of property.
If you want to try to make a reservation, this is where you can expect some work since you will have to interact with the property directly or try to make an online booking and see how far you can get.
Special Note: thx to contributor JonathanReez, I see Booking.com has a Book without credit card option. The result is very limited, but that's an option if you really want to pre-plan.
But, you may not need an actual reservation, especially if you won't know until day of arrival if you need it. In that case, make a list of your preferred properties and if you need the room, call ahead to check availability and ask them to hold a room for you as a walk-in.