Short version: if you are an international user and pick English as your forum language (in your profile), phpBB serves all pages as ISO 8859-1 and cyrillic characters render as garbage unless you change your browser's settings.
Long version:
Anyway, the problem is/was this: all cyrillic characters in the forum show up as garbage characters to me. Actively setting my browser to interpret the forum as a Russian character set worked perfectly, but for a friend of mine it always worked right from the start. His browser picked the right character set all the time, every time. But not mine.
Today I edited my profile a bit, and realized the cause: I've selected to have the forum in English. That's good, since it enables me to read the menu options

And boom!
Garbage characters instead of nice cyrillic ones.