I have to agree with Kevin. I don't see what's wrong with running a trace either, but if that's the edict you have to live with, so be it. You can either try to get the boss to change his mind or live with it.

I use SQL Sentry to monitor my servers. It shows a bunch of graphs and you can go back in history to answer the question "We got an email from some user on the other side of the planet. What happened at 3:42 AM today?" You can identify bottlenecks and go down to the level of what SQL was running at the time. It does the job.

You can also use the DMVs yourself to look for poor performing queries.