How to easily round corners on non-rectangle shapes?

(CS6)

Illustrator's Effect > Stylize > Round Corners effect gives strange results on non-rectangles. How can you take an irregular shape that has sharp corners, for example:

...and round the corners on each curved segment, like I've done manually on the first and last segments here?

Things I've tried:

• Doing it manually with the pen tool takes a very long time. It's difficult to make the rounding consistent or uniform, and it's very easy to accidentally distort the rest of the curves.


• Effect > Stylize > Round Corners gives strange, inconsistent results:

• Inner strokes set to round corners seem to always come out square: