Out of curiosity as an American looking to learn more about European cinema, which countries in continental Europe would you say produce the best movies?
Frankly, I don't know which ones produce the best movies in general as this is somewhat subjective. There are gems from all countries and often there is a co-operation between countries.
However, since I especially like Swedish crime drama, I can give you specific recommendations of crime drama movies and TV-series from Sweden and other countries.
* Snabba Cash I, II, III ("Easy Money" I, II, III)
* Wallander (yeah, the original Swedish one - there are many)
* Beck (there are many)
* Jägarna ("The Hunters", there's two)
* Midnight Sun ("Midnattssol"; TV series taking place in Lappland)
The latter is a French-Swedish production which is, I think, a good example of what the directive of the article intends to create more.
In addition to watching a crime drama of the kind which you might not have not seen, you get to sight-see places as they are. I especially liked the visual appearance of The Midnattssol since it takes place in Lappland, which I think is beautiful in a rugged way.
Also, you may have seen Hollywood remake of The Girl with the Dragon Tattoo. There's a Swedish original which can be interesting to watch.
A Swedish-Danish series called "The Bridge" (Bro) is very good, but I have not seen it.
An Italian (actually Italian-German) series called "Gomorrah" is very good too, I heard. I have not seen that either, though.
I have yet to fully explore all the crime drama from other European countries; I've seen some Finnish ones (like "Vares") and they were OK too.
Not a direct answer, but for TV (serialized crime/police) shows it's the Scandinavian countries. Then the US comes and makes a usually far inferior copy of it.