I would guess that you really want to use requires in your role to require certain functionality of the consumer of your role (in this case Greeting) possibly combined with normal inheritance. For example:

package Hello;

use Moose::Role;
use namespace::autoclean;

# Insist that someone who "with"s us, has a greet method
requires "greet";


sub hello_world {
    my $self = shift;
    $self->greet("hello world\n");
}

1;
[download]

Greeting:

package Greeting;

use Moose;
use namespace::autoclean;

with 'Hello';

sub greet {
    my ($self, @stuff) = @_;
    print @stuff;
}

__PACKAGE__->meta->make_immutable;
1;
[download]

Greeting2:

package Greeting2;

use Moose;
use namespace::autoclean;

with 'Hello';

sub greet {
    my ($self, @stuff) = @_;
    print map uc($_), @stuff;
}

__PACKAGE__->meta->make_immutable;
1;
[download]

Greeting3 (uses the greet from Greeting):

package Greeting3;

use Moose;
use namespace::autoclean;

extends 'Greeting';
# already "with" Hello

__PACKAGE__->meta->make_immutable;
1;
[download]

Now your test script produces:

Greeting=HASH(0x2b411f8)        does Hello
hello world

Greeting2=HASH(0x1a541f8)       does Hello
HELLO WORLD

Greeting3=HASH(0x15c01f8)       does Hello
hello world
[download]

Update: Added script output. Also, I second tospo's suggestion to move all of the common functionality to the Hello role