I assume you are using MyISAM tables? You're pretty much stuck with doing the checks yourself.
So if tableB depends on tableA, then before you delete from tableA you have to delete from tableB (or perform other update) first. With inserts you'd create the record in the main table and then create the record in the dependant table.
It's cumbersome and the only reason I've ever found for keeping a MyISAM table was FULLTEXT indexing. If I knew more about your db restrictions I could possibly make other suggestions.
ETA:
If he's restricting you from using InnoDB tables, I'd wonder about your DBA. In any event, if he won't let you create and use Innodb tables for security reasons, it's really unlikely he'll let you use stored procedures or triggers. So you'll essentially be stuck writing an application to do your CRUD operations in some scripting language.
Therefore you need to think about what the restrictions on these operations have to be for each specific case. Probably the easiest way is to map out your db schema as though it did enforce referential integrity. Include details on what, if any, actions may happen due to a change in any table. Operation sequelae options are RESTRICT, SET NULL and CASCADE.
Once you know how your db ought to respond, you can programs you queries accordingly.
So if Employees have Addresses, and addresses should disappear when an Employee is deleted:
Innodb version (where Addresses has foreign key for Employees, and ON DELETE CASCADE
action)
DELETE FROM Employees WHERE employee_id=7;
MyISAM version:
DELETE FROM Employees WHERE employee_id=7;
DELETE FROM Addresses WHERE employee_id=7;
I hope this makes things a bit clearer.