Load Spikes Every Morning, Every 30 Minutes, Host Can't Solve

We've had a VPS for just over a month now. I am not going to mention the host by name (yet) but they advertise here and other people here reported liking them.

Sadly it's not my experience and I regret my purchase.

Every morning for the past few weeks, we get load spikes every 30 minutes that make our site unusable for a minute (on our VPS, any load over 1.0 is sluggish, over 2.0 is virtually unusable, over 3 is unresponsive)

Here's a series of days as an example:
[url]

The worst part about this is the host insists 1. either it's not happening or 2. they can't find it

I know it's happening because when I try to load a page on the half-hour, it takes over 13 seconds (less than 1 second normally). And it's fairly obvious it's someone doing a cron job with some nasty downloading, uploading, or maybe a massive mysql update.

Someone tell me what to tell them because this is driving me out of my mind. The load is NOT being caused by ourselves, I've made sure all our cron jobs don't happen on the exact half hour and we get lots of traffic later in the day without loads.