There are occasional reports of problems in situations when there is no employee to deal with (such as at unattended gas pumps). There can be an issue there, because there's no way to collect your signature. In other situations, you'll often be asked to sign if you don't tap to pay, and even with tapping I think there may be an upper limit, above which you'll be asked to sign.
I've spent a huge amount of time in Europe since 2015, in 14 different countries, and I've only once been unable to complete a credit card transaction. I was trying to buy a bus ticket at a little vending machine on the street in Warsaw. Having US credit cards without PINs for purchases hasn't been a significant issue for me. Cards are different, so I cannot guarantee you will run into zero problems, but I think you will probably have no difficulties when you are dealing with a person as opposed to a machine. Some folks have said they got around a machine prompt for a PIN (when no employee was present) by entering 0000 or some random 4-digit number. I can't vouch for that, but I don't think it would hurt to try it.