which credit cards accepted along the Rhine

IME, VISA and MC are widely accepted, Discover not so much.

Discover is rarely excepted outside of the United States. Some of them have a MasterCard emblem on them and will be accepted but a lot of times merchants won’t take them because they don’t know that. I would just leave it at home.

A lot of places in Germany are either cash only or cash strongly preferred (meaning they will tell you they don't take credit cards), so have cash, too.

I wouldn't say a lot of places in Germany don't accept credit cards. I spent a month there in 2022, including the Rhine area, and there were only a couple of places that were cash only. But I do agree it's good to have a bit of cash with you.

Figuring where the cruises stop, and the places passengers have access to, a card, with contactless capability (nearly all cards now) or your phone with Google/Apple Pay should be all you need. Yes, having some euros is a must, as back-up, but if you get by without needing them, spend them at the end of the trip.

I say that though I am in Germany now, and using a good amount of cash, more than I have on the last 4-5 trips combined. But I am in smaller towns, places like cash, went to a wine festival, maybe could have used card, but no one else was, and I did not want the looks if I repeatedly tried to use a card for 3 euro wine pours. The regional buses only take cash, unless you use the app for tickets, and there have been other examples, but nothing you would run into in the main tourist centers.

A lot more places take credit cards these days, but the small places I stay at and most places I eat at only take cash.

I’ll just add that even if a small business accepts credit cards, there may be a required minimum purchase of between €5-€15. Different shops have differing amounts.

"...using a good amount of cash...." Very true. That applies to this summer too. Both small and big places want only cash in both tourist frequented areas and in non-tourist areas.

I would not assume that in a big establishment you have that credit card choice in Germany. Maybe more likely but not necessarily so. I have seen in small eateries Americans presenting a cc only to be told they had to go the ATM to get cash. That was in Berlin and twice in restaurants in Vienna.

unlikely you will be able to use the Discover anywhere in those three countries, except perhaps an American chain hotel, and likely not there.

In all my trips to those three countries since covid I have spent just about 10€ in cash, the rest all tap and pay. (Also Luxembourg and Belgium).

I do enjoy the CC versus cash discussion.
Cash is still quite popular even in a big city like Frankfurt. The cafe I go to every morning downtown is cash only. The eis salon that I go to at least once a week for my spagetti eis, is cash only and the elegant Georgian restaurant that we went to last week, was also cash only. I was surprised by that one.
Once you move away from the tourist areas, neighborhood restaurants, small food truck stands, farmers markets in small towns or local festivals, be prepared to pay cash.

Never a bad idea to ask about card acceptance before placing one's order.

"Never a bad idea to ask about card acceptance before placing one's order."

I would say that in Germany, it is a very bad idea NOT to ask before placing one's order. I don't typically carry cash at home. On our first night in Berlin, I had to go back to our apartment to get cash to pay for our dinner. Thankfully, we were only staying a block away. This was in 2018 and in a non-tourist area. I've heard it's changed somewhat since then, but on my recent visit to Wiesbaden and the Mosel Valley, there were definitely places that were cash-only.

Some other ways to check prior to actually placing your order are to see if the MC/Visa decal is on the window of the entrance door, or if the menu usually on the last page indicates cc payment is accepted. Then there are those restaurants that make it explicitly clear what the policy regarding payment with their "Cash Only" sign on the front door.

Failing all that, then ask. I have seen restaurant patrons, not just Americans, be told when paying that the cc was not accepted,

at which point some had to leave after being informed the location of the closest ATM to get the cash or just started digging in their wallets to come up with it. They all viewed this as an unexpected and unpleasant surprise in Germany and Austria.

Visa and Mastercard for sure. American Express is accepted in large hotels or places linked to the USA, but not as widely accepted.

Be sure that your card scan feature works.