You could use Email::MIME, but I think you'll somehow have to handle the separate cases...  As multipart messages can in theory be arbitrarily nested, a recursive approach is appropriate.  The following example returns the first acceptable (i.e. plain text or html) part found:

use Email::MIME;

sub handle_parts {
    my $part = shift;
    my $content_type = $part->content_type;
    #print "Content-Type: $content_type\n";  # debug

    my $body = $part->body;
    
    if ($content_type =~ m#text/plain#) {
        return $body;
        
    } elsif ($content_type =~ m#text/html#) {
        return html2text($body);
        
    } elsif ($content_type =~ m#multipart/#) {
        for my $subpart ($part->parts) {
            my $text = handle_parts($subpart);
            return $text if defined $text;
        }
    }
    return;
}

sub html2text {
    my $html = shift;
    # my $text = ...  (left as an exercise)
    # return $text;
    return $html;
}

my $message = ...

my $parsed = Email::MIME->new($message);

my $text = handle_parts($parsed);
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You might need some additional checks to figure out if the first content part is the one you want...

Thanks! This approach worked quite well. It seems like it would be a useful function to add to one of the many email modules on CPAN