Mastercard and Visa vs American Express

There are plenty of cards with no foreign transaction fees - you can get any credit card you qualify for (regardless where you bank) just by applying for it. American Express is not as widely accepted as Visa/Mastercard.
You can compare credit cards on this site: www.bankrate.com

American Express fees cost merchants in the U.S. quite a bit more than Visa or Mastercard, and I assume that would be the case in Europe too-and perhaps why it is not as widely accepted, as Agnes says.

If a business (shop, hotel, restaurant) accepts credit cards, they will accept Visa and Mastercard.
If they accept credit cards, they will usually accept visa and Mastercard debit cards, which are used as much as credit cards in Europe.
American Express is a lot less widely accepted, chiefly because they charge the business more, and very few people in Europe have them.
I have never heard of discover card, but a bit of googling tells me it is a US-specific brand, i.e. nobody has heard of it in Europe.

Discover seems to have a relationship with Diners Club. Neither of them are widely accepted outside the US. Stick with Visa and Mastercard (which has a different name overseas, perhaps Maestro?), use Am Ex as a backup. Have an assortment of both kinds from different banks spread among your travel party. At some point in your trip, for no apparent reason, one card will not work somewhere but the others will.

On 2014 tour Germany, Austria, Switzerland, we asked at just about every shop and restaurant we went to, if they took AmEx. I'd say only 2-3 out of 20+ did. I'm not talking about small mom & pop shops either.

However , a good reason to take your Amex is for auto rental if you sign up in advance for the extra Amex insurance (charged by the rental, not by the day as the rental agency does.)
That said, I use my United Mileage Plus Explorer card from Chase to rack up miles without FX charges. Also has primary coverage for auto rental insurance.

I agree that it would be better to have a Visa/MC. AMEX isn't accepted very many places, especially the small, independent restaurants and cafes that appeal to me.

However, you could get the AMEX and use it for hotels. I think a lot of hotels take AMEX.

There are lots of cards out there with no foreign transaction fees. Some have no annual fees and some have rewards. Check out BankAmericard Travel Rewards. Also check out Barclay Arrival+ and Chase Sapphire Preferred, but those do have annual fees. Airline co-branded cards like the United MileagePlus card from Chase could be another good option.

I agree with most of the above. Visa and Mastercard are accepted nearly everywhere as they have European equivalents in the same network. American Express is less accepted, especially 10 years ago, but with expansion under Delta Gold Card and other retailers plus the move to a conventional revolving credit scheme as opposed to pay in full monthly plan, acceptance in Europe has increased greatly. Hotels, major transport options and many tourist oriented businesses will likely accept AE. For the smaller places, paying cash from an ATM (with no fees) is still the best option. If you have the time and option to get a second card from Visa or MC that has no foreign transaction fees, that would be a prudent step as well.

Visa and Mastercards are better accepted, but all Visa's and Mastercards are not all created equal.

I use a Capital One Visa because I'm getting travel "points" used to travel with. But they also pay the full value on the day's currency conversion--something not all credit cards do. Many charge cards "discount" their cards adding on a small percentage for usage.
We use the credit card any time I can use it--rooms, restaurants, etc. For cash, I use my credit union's ATM card with no up charges. My Wells Fargo checking ATM charges $5 plus 3% discount every time I use it. And I don't carry any appreciable amount of cash when traveling.
I prefer to stay away from American Express as I have no need for another credit card.

American Express used to have offices everywhere and was the traveler's choice because of the level of service they provided. But no more. Most of their travel offices in Europe have closed. Their higher fees and stricter membership rules for the merchants has resulted in less acceptance in Europe. Your best bet is a Visa or MasterCard (or one of each so you have backup if one quits working). Discover, no matter what their marketing pitch tells you, is practically unusable outside the US.

I would suggest getting a Capital One Visa or MasterCard as your primary credit card when traveling to Europe. No fees of any kind and they charge the Google rate for foreign exchange. Meaning if Google says 1 EUR = 1.09USD, that's what you pay and not a penny more on your transactions. (Standard interest charges apply and there are fees for getting cash out of ATMs with their credit cards, see disclosures for details.)

Another branch of Capital One called Capital One 360 also has a MasterCard debit card with zero fees which means as for the credit card what you see on Google as your rate is exactly what you are charged. In fact, their entire line of banking products have zero fees. You can open an account with them with no minimum opening amount and no minimum balance requirements. They also allow free transfers into the checking account to fund the debit card. I have used them for over 10 years in my travels and have never had any problems with the card and being able to get cash when I needed.

I use my Capitol One 360 debit Mastercard for cash withdrawals throughout Europe without any issues at all anywhere. I got it specifically for traveling when I found out my "regular" bank Visa debit card would charge me a 3% foreign transaction fee. Hotel fees and large purchases go on my Barclay Arrival Mastercard to earn points. Visa and Mastercard (debit or credit) are accepted almost everywhere, AmEx not nearly as much.

When we were in France a couple of years ago, American Express wasn't accepted. (found this out before our trip but verified it when we saw restaurants refusing others' AmEx cards.)

We pay hotels with Visa (if I haven't already paid ahead of time) and use our ATM card for cash all other transactions.

"There are lots of cards out there with no foreign transaction fees. Some have no annual fees and some have rewards."

Capital One Quicksilver is one of them, and my primary credit card for travel. But I don't use it, or any other credit card, to withdraw cash from ATM.

http://traveltips.usatoday.com/avoid-credit-card-atm-fees-traveling-abroad-103405.html

"Avoid using your credit card for cash advances overseas, which typically accrue not only fees but also finance charges until you pay your bill in full."

For cash withdrawal from ATM, I use an American Express debit card issued by my investment firm. All ATM fees are refunded in full. Cards like this are much harder to come by.

Norwegian, you asked only about "brand". Can we assume your cards have a chip already? You should also consider that even MC V card acceptance is not as universal in Euope as it is, say, in the U.S. Where do you live?

"For cash withdrawal from ATM, I use an American Express debit card issued by my investment firm. All ATM fees are refunded in full. Cards like this are much harder to come by." If you are not married to an American Express debit card, there are many other debit cards with no transaction fees and at least Schwab's VISA debit card refunds any fees charged by an issuing ATM (as the ones near me do)...easy, easy to get.

A brief Internet search this morn revealed that VISA and Mastercard charge very similar but not the same rates when converting purchases in euro to US dollars. The AMEX conversion rate was not found.

Discover is making some headway in Europe, but slowly. The relationship with Diners Club works sometimes but not always.

When booking hotel rooms in advance, I use Amex to secure the reservation but pay with Visa, MC or Discover.