I'm both a car driver and a cyclist. There are a zillion roundabouts in the country I'm living in. This is my experience and I'd like to hear a traffic designer about it.
Roundabouts feel safer than a 4 way intersecting when I'm in a car even if it's annoying to have to slow down and accelerate again when there is very little or no traffic, which is normal in the countryside. It also consumes more fuel, wears the car, adds to pollution.
It's different when I'm riding my bicycle. Roundabouts put me closer to cars, especially if there are traffic dividers that force cars closer to the side of the road. Then cars are usually slower than a bicycle in a small or medium sized roundabout (think of race bicycles with drop bars,) which adds up to chances of being packed between two cars. My feeling is that what's good for cars is bad for bicycles.
As a cyclist I can see what's around me much better than when I'm in my car. Intersections don't feel like a problem because of the extra visibility. When I'm riding I long for old times straight roads, with a line on the side and half a meter of asphalt on the other side to ride on. No roundabouts, no speed bumps especially the 3M style plastic ones, which I can jump but they are annoying. All those safety devices, including roundabouts, increase fatigue and danger. They are also the main reason for crashes in bicycle races but that's not an issue for normal people.
Finally, as I both drive and ride I know how to behave not to put in danger the cyclist. When I drive I don't overtake bicycles before a roundabout because they'll be at least as fast as me. I don't overtake when there is a short traffic divider which narrows the road. A long one, oh well, I have to. Same thing when I approach an intersection where I have to yield or stop. Bicycles will be definitely faster there because they can look around better than me and decide what to do quickly.
When I ride and I'm approaching a roundabout or entering a short narrowing of the road I look behind me and if it's safe I lean a little to the center of the road to prevent cars to overtake when they'll be forced to come close to me.