CPU usage

I have read numerous threads having to do with excessive CPU usage on shared servers but I don’t see any consensus about how the issue can be best addressed.

Starting a week ago, my provider, InMotion Hosting has been suspending my account, parterre.com, on a daily or more-than-daily basis. Briefly, the host says a) “your CPU spikes started about half way though last month” [i.e., April 2009] and b) “we are still seeing excessive use of CPU and RAM, which may be related to a high number of MySQL queries being issued by your wordpress installation.”

At the time the spikes started, I was running 2.6 and had been doing so for about a year (?). I had not made changes in frequency of postings or the plugins any time in the past six months or so.

Since the first suspensions, I have installed 2.7 and activated WP Super Cache. Meanwhile, usage spikes continue, most recently sometime between midnight and 7 am today, a time when there is practically no traffic to my site.

Any ideas? I apologize in advance for not being able to supply more information, but with the site suspended I have no access to stats or even my control panel.