Here's the code:

#!/usr/bin/perl -w

use MIME::Parser;
use DBI;
use strict;

my $database = "xxxxxxx";
my $dbuser = "xxxxxxx";
my $dbpasswd = "xxxxxxx";
my $outputdir = "/some/directory";

my $parser = new MIME::Parser;
$parser->output_dir($outputdir);
$parser->output_prefix("attachment");
$parser->output_to_core();
my $entity = $parser->read(\*STDIN);
my $msg_header = $entity->stringify_header;
my $num_parts = $entity->parts;
my $msg_body;

if ($num_parts > 0) {
    for (my $i = 0; $i < $num_parts; $i++) {
        my $part = $entity->parts($i);
        my $type = $part->mime_type;
        my $bh = $part->bodyhandle;
        if ($type =~ m/text/) {
            if (my $io = $part->open("r")) {
                while (defined($_ = $io->getline)) {
                    $msg_body .= $_;
                }
                $msg_body .= "\n\n";
                $io->close;
            }
            my $status = system("rm '$bh->{MB_Path}'");
            die $! unless $status == 0;
        } else {
            my $file = $bh->{MB_Path};
            my $oldfile = $file;
            $file =~ s/$outputdir\///;
            my $now = time();
            $file = $now . "-" . $file;
            my $newfile = $outputdir . "/" . $file;
            my $status = system("mv '$oldfile' '$newfile'");
            die $! unless $status == 0;
            $file = "some/directory" . $file;
            $file = "http://url/" . $file;
            $msg_body .= "ATTACHMENT: $file\n";
        }
    }
} else {
    if (my $io = $entity->open("r")) {
        while (defined($_ = $io->getline)) { 
            $msg_body .=  $_;
        }
        $msg_body .= "\n";
        $io->close;
    }
    $entity->purge;
}

### The rest is just database processing ...
[download]

The problem is that the parser dies on some mails when it should remove those temporary files.

(I know this is dirty code, but I made it most out of trial-and-error.)