I think the largest problem with roundabouts in cities is that crossings are often 'linked' to each other. If you have a traffic light on one crossing, and a roundabout on the next, then the traffic towards and from the roundabout is influenced by the traffic light. This is especially a problem if the crossings are close to one another.
In The Netherlands there has been a large increase in roundabouts, since 15-20 years. You see them virtually everywhere, even on 100km/h roads. The only objection so far is that emergency services, like ambulances and firetrucks have more difficulty passing other cars on roundabouts than on traffic lights, because the emergency services can manipulate the traffic lights, but not the roundabouts of course.
The approach that if situations are more difficult, people will start paying attention is also used in the "Shared space" concept. In The Netherlands this concept became quite popular, especially by Hans Monderman, who implemented it in for example the city of Drachten as well the village of Makkinga. In the latter they removed _all_ roadsigns, road markings, stopping restrictions, parking restrictions, etc. In Drachten they replaced a really busy crossing (22,000+ cars/day) with a roundabout which works really well. Searching for videos on 'shared space Monderman' gives some really interesting results, I would say.
The idea of Shared Space is that a lot of road signs are removed, making people actually look to each other (making eye contact to see what the other person is going to do) instead of "looking at the lines" ("coloring between the lines"). This is actually only possible if your driver license is sufficiently difficult to get, so people not only know how to drive, but also learn how to anticipate to potentially unexpected events.