Here are two versions: the first one intermixes all the lines from the files, the second allows you to decide on a limit of how many lines to read from the files (as per your followup specifications):

#!/usr/bin/perl -w
use strict;
use IO::File;
my @files = qw/file1 file2 file3/;
my @fhs = map{IO::File->new($_)||die "$_: $!"} @files;
print while $_ = join '', map{scalar <$_>||''} @fhs;
__END__


#!/usr/bin/perl -w
use strict;
use IO::File;
my $limit = 2000;
my @files = qw/file1 file2 file3/;
my @fhs = map{IO::File->new($_)||die "$_: $!"} @files;
for(1 .. $limit) {
    $_ = join '', map{scalar <$_>||''} @fhs or last;
    print;
}
__END__
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Both assume that you'll just redirect output to the new file, but you can open() an output filehandle and print to it if you wish.