I agree it's not strictly religion. The Roma appear to be just as despised as the Muslims currently are. It used to be the Jews who didn't "assimilate" with the white Christians. Many of them probably still don't in the eyes of the anti-Semites.
Roma or Gypsies, in the local image, tends to strongly refuse integration, harbors large percentage of criminality, but at the same time pretends a lot from the state (in terms of funds, etc.).
It's typical that they do not respect age limits for children work, and refuse to send their children to state schools (this is against the law in Europe).
Long story made short: their behavior strongly contrasts with local laws and expectations; at least in the eyes of citizens of many European countries.
Again it is not about race and religion.
In other words, keep a low profile and don't provoke the local rednecks?
No.
Actually it's more about: if you behave antagonistically against local people, don't be surprised if they get pissed at you.
I think that is utter nonsense. Most, if not all, of these countries actually had more crime before Muslims started immigrating in larger numbers. Crime in general has decreased over time. Not the opposite.
In absolute terms you may be right (I don't have stats to confirm/deny your statement that I will take as true).
What we have observed in Europe, according to the stats that once in a while get published, is that the incidence of crime in specific immigrants communities has grown very fast.
Another way to see it is that some minorities are a majority... and this is not about having more air-time on TV but about facts.
This is not about religion and not about race and colour.
It's about poverty, cultural background, and low education of many of the new immigrants, that have taken the place of the lowest "classes" in many European countries.
European citizens are tired of a long time of empty promises about integration.
They see, right or wrong they may be, that a lot of troubles come from immigrants and the failure to integrate (assimilate) them.
European politicians have ignored the issue for long time, and now the problems have become bigger and the mutual incomprehension bigger as well.
It's a problem of perception as well, and I admit that sometime politicians exasperate the terms to gain more votes... but ignoring the problem is not the solution either.
And, as others have pointed out, it is largely a class issue. Poor people tend to commit more blue-collar crime than rich people do. The latter typically prefers white-collar crime which goes largely unpunished.
That's what I stated too.

One important point is that what you call "blue-collar crime" is felt mush more strongly by people.
A high-level crime (e.g. accountancy fraud) is felt very far from personal experience.
A robbery in the house is felt much more personally and extremely damaging.
For this reason people tend to be more angry against the "poor" robber than the jet-set robber.
Ignoring this small fact and "forgive" the poor robbers doesn't help.
This is what European politicians have done for quite some time and now they have to show themselves strong against criminality to don't loose votes.