You want to look at the /s switch on the s/// operator. For instance, with the following code (in a one-liner with -l):

my $str = q{this is "a test} 
    . qq{\n} 
    . q{this is "only a "poor "test}; 
print $str; 
print qq{=====}; 
$str =~ s/"[^"]+"//gs; 
print $str;
[download]

this is "a test
this is "only a "poor "test
=====
this is only a test
[download]

Hope that helps.

Update: 2011-03-09

I cannot help but second roboticus's references and suggestions. There is much useful information (and I now have another module to take a look at when I have "spare" time available again). :)

Update: 2011-03-09

It appears as if the module in question may be YAPE::Regex::Explain (as I found when I tried installing it to play with it myself).