Yeah, you have path issues, and possibly regex issues. mv requires full path names. The following works. In testdir1 I have 2 subdirs Results_2008 and Results_2009. In those subdirs , I have files:

Results_2008:
2008_12112L5_LacZ  2008_12112L5_pgK  2008_12112L5_SD  2008_12112L5_SU

Results_2009:
2009_12112L5_LacZ  2009_12112L5_pgK  2009_12112L5_SD  2009_12112L5_SU
[download]

this script moves them all, and notice the regex(probably can be improved) and full path in the system mv

#!/usr/bin/perl
use warnings;
use strict;

my $topdir = "/home/zentara/testdir1";
my @subdirs = get_sub_dirs($topdir);
my $newdir = "/home/zentara/testdir2";

sub get_sub_dirs {
    my $dir = shift;
    opendir my $dh, $dir or die "Error: $!";
#    print "$dh\n";  #this is your glob
    my @dirs =  readdir $dh;
    @dirs = grep /^Results(.*)$/,@dirs;
    closedir $dh;
    return @dirs;
}

print "subdirs = @subdirs\n";

foreach my $dir(@subdirs){
    opendir my $dh, "$topdir/$dir" or die "Error: $!";
    my @files = readdir $dh;
    print "raw files = @files\n";
    @files = grep /^[\d]+_[\dL]+_(LacZ|pgK|SD|SU)(.*)$/,@files;
    print "$dir -> @files\n";

    foreach my $file(@files){
     system( "mv $topdir/$dir/$file $newdir/$file");
      
    }
    closedir $dh;
}
[download]

Also remember, if you want to separate your files out into subdirs, you need to make the subdirs first, before trying to mv them there. See cloning a directory tree