DELETE data from a table, joining through two tables

You need to delete rows from tt, not individual columns:

DELETE tt  FROM ItemTracker_dbo.Transaction_Type tt  JOIN ItemTracker_dbo.Transaction t ON tt.Transaction_ID = t.Transaction_ID  JOIN ItemTracker_dbo.Purchase p ON p.Transaction_ID = tt.Transaction_ID  JOIN ItemTracker_dbo.Item i ON i.Item_ID = p.Item_ID WHERE i.Client_ID = 1

The syntax is incorrect - you don't reference columns between the DELETE and FROM. Use:

DELETE FROM ItemTracker_dbo.Transaction_Type tt  JOIN ItemTracker_dbo.Transaction t ON tt.Transaction_ID = t.Transaction_ID  JOIN ItemTracker_dbo.Purchase p ON p.Transaction_ID = tt.Transaction_ID  JOIN ItemTracker_dbo.Item i ON i.Item_ID = p.Item_ID WHERE i.Client_ID = 1

To be sure you're deleting the correct stuff, I agree with wallyk that you should check the output of the SELECT statement that what is returned is what you want to remove prior to. Otherwise, perform the delete in a transaction so you can roll it back if needed.

You should compose it initially as a query to return the rows of interest. Once that's all debugged, then convert it into a delete.