In the past I’ve had little trouble using my magstripe-based American credit cards in Europe. However, these have now been (finally!) replaced with EMV cards to catch up with the rest of the world. Unlike the rest of the world, my new cards are set up for chip-and-signature rather than chip-and-PIN. Will these work on European EMV terminals that are set up for PINs? Are they smart enough to prompt be for a signature instead, or do I have to somehow set up my credit cards with a PIN?
The readers that restaurants use will happily print out a slip to sign.
Kiosks (subway card vendors, for example) won’t do that, but they seem to work without a PIN or signature (at least, I didn’t have any problem in London last year).
Last year I couldn’t get my AmEx to work in a ticket machine for the Zurich tram system, but IIRC that was still the old magstripe version. Fortunately I had enough primitive metallic currency discs to buy a fare.
I’m having a surprising amount of trouble finding a straightforward answer for this using my googler.
I got a PIN prior to taking my new credit card to Europe. In general, the card readers recognized that this card wanted a signature, not a PIN. Several merchants were surprised by this, but gave me a piece of paper to sign. Kiosks required me to enter my PIN.
It was much easier than my last trip, when many machines refused to handle my old mag-stripe card at all. When I bought a pass for the London tube, a clerk had to go dig out an old machine from the back to cope with my card. The mag stripe caused lots of delays like that, and sometimes was just unusable.
On this trip, a friend who still has a mag stripe did get a nice meal for free, because the restaurant couldn’t process his card, he didn’t have enough cash, and the manager decided it was easier to just comp him than to find some other way to extract the fee from him. 
Yes.
The terminals know the card is a signature card and will print out a receipt. Some clerks were surprised when it happened, but I didn’t have any trouble. This worked at restaurants, grocery stores, train stations, and all other retail stores I visited.
Some unattended kiosks in isolated locations won’t work. In Germany, these were things like bus stop ticket kiosks.
Cite: my trip to Germany in early June this year.
I was in Germany and Denmark, in July of this year. My London trip was August 2014.
The reason you cannot find a definitive answer is because there isn’t one. I hear anecdotally that ATMs will almost always accept most cards. Nearly all larger retail premises (supermarkets) will accept cards without chip-and-pin because there was some concern about the elderly and disabled not coping.
The main problems seem to be with ticket machines at stations and bus stops and at petrol stations at night.
As an aside, in the UK at least, cheques are pretty much obsolete and many shops will not accept them.