Book (by default) and report (with openright class
option) ensure that each chapter starts on a right-hand (recto) page;
they do this by inserting a \cleardoublepage command between
chapters (rather than a mere \clearpage). The empty page thus
created gets to have a normal running header, which some people don't
like.
The (excellent) fancyhdr manual covers this issue, basically
advising the creation of a command \clearemptydoublepage:
\let\origdoublepage\cleardoublepage
\newcommand{\clearemptydoublepage}{%
\clearpage
{\pagestyle{empty}\origdoublepage}%
}
The "obvious" thing is then to use this command to replace
\cleardoublepage in a patched version of the chapter command.
(Make a package of your own containing a copy
of the command out of the class.) This isn't particularly difficult,
but you can instead simply subvert \cleardoublepage (which isn't
often used elsewhere):
\let\cleardoublepage\clearemptydoublepageNote: this command works because
\clearemptydoublepage uses a copy
of \cleardoublepage: see
patching techniques for explanation.
Note that the KOMA-Script replacement for the book class (scrbook) offers class options that control the appearance of these empty pages, and Peter Wilson's memoir class has similar facilities.