how are Band B's reserved in advance if they are "cash only"?

That depends entirely on the particular b&b. You need to contact them or check their website (if they have one). Some require a cc# to reserve and some require a cash deposit, sometimes by paypal or money order or bank transfer.

I have two reservations for B&B's one in Haarlem, Netherlands and Bayeux, France. Both of these are cash only. The reservations are made through e-mails with the owners. They trust you will keep your reservation. They will want you to pay for the nights (I believe this anyway) when you arrive in cash. The two owners have been very helpful and nice whenever I have contacted them. They responded very quickly also. They will want you to contact them just before you arrive to ensure that you are keeping the reservation. Mine are for April 17-18 and May 9-10, 2015. I will let you know how it goes when I return if your travel plans are after these dates.

Most likely for the reservation they will ask for a good credit card, which they will only process payment if you do not show. Possibly they will have you make a deposit through Paypal. If you deal directly, you will find this out via their reservation process. if you are dealing with a booking service, the service is almost certainly going to do this via credit card (and this will likely mean the place is making less money as they are paying a fee to the booking service)

Some cash-only places will take credit card info to hold a reservation for you, but they never want to 'run' the card. Sometimes they are true dinosaurs and really have no system at all to accept electronic payment.

If your research has you confident, through crowd-source reviews, etc., that the places you want to stay are trustworthy. You should definitely stay with them.

I've never had any sort of problem when the owner of a small establishment takes only my basic info and gives their word they will save a room.

We stayed at La Grange Chambre last summer. We gave the owner our credit card number to hold the room and then paid in cash once we were in Amboise. Yveline is a wonderful hostess!

Thanks to all of you for your answers. The info will help us decide.