$query = "
SELECT *
FROM ITEM
INNER JOIN DIRECTORYITEM ON (ITEM.ID = DIRECTORYITEM.ITEMID)
INNER JOIN DIRECTORY ON (ITEM.DIRID = DIRECTORY.ID)
WHERE DIRECTORY.ID='$prodtype'
";
That is your query written correctly. In my original example when I wrote "INNER JOIN table2 t2" the t2 in that statement was an alias for the table (just used so I didn't have to type out Table2.id), it looks like you were trying to identify the field to do the joining on in your querys. So I removed those entries, plus I made your ON clauses read correctly. When you do a JOIN the ON clause tells the database how you are JOINing these tables together so you need to populate it with the correct fields from databases that you've already identified in your query. So on your line "INNER JOIN DIRECTORYITEM" you have to make the ON clause tell the database how DIRECTORYITEM and ITEM relate to each other.
Finally your WHERE clause was incorrect, you don't separately quote the tablename and fieldname.