How to delete all rows in a Greenplum row oriented table

You're interpreting the semantics of the delete statement incorrectly. When a using clause is used, it doesn't mean that records will also be deleted from those tables. Instead, those tables are purely used to join to in order to determine which rows need to be deleted from Users.

You basically have three choices:

1. deleting child rows in a before delete on Users trigger.
2. on delete cascade constraints.
3. execute multiple delete statements on the various tables involved, in the right order.

My preference, in certain cases, is actually for the on delete cascade constraints, but I don't use them everywhere: just for the situation where it makes sense to be able to remove all of the children of a given parent in one go. I might use it for "invoices" and "invoice_lines".

When you take this approach you need to be sure that only users who really need to be able to delete from the parent table have that privilege -- no users or applications logging in as table owners!

Quoting the manual:

There are two ways to delete rows in a table using information contained in other tables in the database: using sub-selects, or specifying additional tables in the USING clause. Which technique is more appropriate depends on the specific circumstances.

Bold emphasis mine. Using information that is not contained in another table is a tad bit tricky, but there are easy solutions. From the arsenal of standard techniques to ...

• Select rows which are not present in other table

... a NOT EXISTS anti-semi-join is probably simplest and most efficient for DELETE:

DELETE FROM link_group lg
WHERE  NOT EXISTS (
   SELECT FROM link_reply lr
   WHERE  lr.which_group = lg.link_group_id
   );

Assuming (since table definitions are not provided) link_group_id as column name for the primary key of link_group.

The technique @Mihai commented works as well (applied correctly):

DELETE FROM link_group lg
USING  link_group      lg1
LEFT   JOIN link_reply lr ON lr.which_group = lg1.link_group_id
WHERE  lg1.link_group_id = lg.link_group_id
AND    lr.which_group IS NULL;

But since the table expression in the USING clause is joined to the target table (lg in the example) with a CROSS JOIN, you need another instance of the same table as stepping stone (lg1 in the example) for the LEFT JOIN, which is less elegant and typically slower.