You had a few simple errors. This works:

use Digest::MD5 qw(md5_hex);
use strict;
use warnings;

my $secret = "applepie";

sub get_md5
{
    md5_hex("$secret" . "$_[0]")
};

my @array     = qw(foo bar baz);
my %md5_hash_broken  = (map {&get_md5($_) => $_} @array);

my %md5_hash = ( &get_md5('foo')  => "foo",
                 &get_md5('bar')  => "bar",
                 &get_md5('fubar')  => "baz",
               );

my ($x, $y);
while (($x, $y) = each %md5_hash)
{
    print "$x => $y\n";
}
[download]

The errors (see my changes):

• I got a warning on the use of $_ in get_md5();
• Your map was not passing an argument to get_md5();
• Your hash called get_md5() with the argument bar twice.