I came across a page here that offers a good explanation of how to use Perl for multiline search and replace. I thought I'd make a note here for future reference.
$ perl -pe 'undef $/; s/stringOnLineOne.*\nstringOnLineTwo//' inputFiles.*
See the section on that page entitled "The Field Record Separators."
A post at perlmonks shows how to use this in practice for multiline search. See the last answer to the question on this page.
For good explanations of the "-p" and "-e" flags, see this page. In brief, "-e" allows entering commands via the command line (as opposed to via a file), and "-p" allows looping over input files and printing the output.
