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I have three flowering (not yet actually flowering) hydrangeas in
containers in my courtyard, which is generally very shady and
sheltered. For the last three years, the hydrangeas seemed to be
growing pretty well, though the one which got the most sun grows less
than those which get more sun. But this summer, about a month ago,
the leaves of the sunniest one started to turn brown, and then the
middle one, too, and now all three have brown leaves. At first I
thought it might be sunburn, but the conditions haven't changed since
they were planted, so why just now? There has been root rot in other
containers in my courtyard, but these leaves are not wilting, just
turning brown.

It is most likely a fungal disease. Here is a like to a wonderful article,
with pictures, on diseases of hydrangeas:
http://www.aces.edu/pubs/docs/A/ANR-1212/ANR-1212.pdf