We had great difficulty with using credit cards in Copenhagen Denmark. Our Debit card saved us. Machines over there require a pin
We have used our Capital One credit card there, without any problem. We have always had success with either the Capital One or getting cash with our Travelex card.
We used our Capital One magnetic strip card in the first taxi to the hotel. No problem. The taxi to the airport would not accept it even though we mentioned the other taxi driver did accept it. His response was "That driver must have been new". Our debit card also saved us.
BTW- the taxi drivers there have limited English skills (not Danish) and even more limited social skills or desire to be nice.
When you say "machines," I wonder if you mean machines that you operate yourself, in which case I wouldn't be too surprised, or any card-processing machine that a store or hotel may be using to process your larger purchases? Also, you can get a pin for a credit card, even if it is not technically the same chip-and-pin system. See also http://www.ricksteves.com/travel-tips/money/chip-pin-cards.
I had one merchant who didn't know how to accept a magnetic strip card. That was in Helsingor at a small shop off the main street. No problem, we walked a block and got cash from an ATM and went back. Other than that one shop, everyone else took my card without a problem.
Just got back from Scandinavia. The rest of the world has implemented a chip called EMV, while most US card issuers have not yet mailed them out. They will come in the next year, as even US must accept them for their enhanced fraud prevention. These also use a PIN like your debit card.
In DK and other Scandinavian countries, I could usually say "no PIN - it requires signature" and they would swipe it and I could sign. However, there were some places, including the train sales office at Copenhagen airport that could not accept my credit card. Good news is I never had an ATM machine that would not take it - so I'd get about $200 in krone for emergencies from the ATM. The ATMs had decent exchange rates and clearly spelled out what the rate was.
Be sure to tell your bank you'll be in these countries! They will need to allow transactions to go thru.
Most restaurants, museums, rail stations, and stores in Scandinavia were able to take my US credit card - but yes, a few did not - including a taxi in OSLO that said it "didn't work" after he swiped it. I then paid cash (luckily I had NOR krone). When I got home, sure enough the swipe had worked and I paid double. Watch the taxis in particular - often shady in all countries.
Some places in Denmark still only accept Dankort and no other type of card.