The quick and dirty argument that should come up against such policy is the implementation of business logic.
Mundane auditing of data traffic is only one good use of triggers.
Maintenance of those triggers for business logic makes the use of triggers take on a life of its own, as if it is part of the software lifecycle. For some databases like Oracle, PostgreSQL and SQL Server, such use of triggers in an ACID compliant arena should never be suppressed or relegated if business logic is best implemented in the database.
On the other hand, one DBMS that has issues with ACID compliance in triggers is MySQL. The stored procedure language in MySQL has a rocky history although a measure of success of implementation can be achieved.
Getting away from any specific database and to keep things in perspective:
If referential integrity is to be maintained, let the database's storage engine do it. Why should an application reinvent the wheel of database integrity as part of business logic ? What if a transaction (series of SQL statements) needs to be rolled back and the triggers have already fired off in the middle of the transaction ? Can the audit trail info that was recorded be rolled back ? Does the storage engine allow rollback for trigger level data ? Will you have to design the rollback features independent of the paradigms of standard SQL ? Due diligence of research on these topics will clarify whether triggers are a healthy alternative.
If there are some arithmetic operations you can perform in a trigger that will not slow down the processing of each row in a table, then triggers are OK.
If there is some massaging of data you need to do and you'd rather have a trigger do it instead of at the browser code (PHP, Python, Ruby, whatever) that will not slow down the processing of each row in a table, then triggers are OK.
IMHO you should make sure audit trails are handled in transactions rather than in triggers because each recording of a row causes intermittency of database operation, especially when doing bulk processing or in a high-trafficked website. Triggers can accomplish business logic in many (but not all) situations.