I didn't see it mentioned before, but has anyone else noticed that the server seems to go AWOL for ~5 minutes at a time every now and then?
Yeah, we know what causes it, and was aware the day before migration.
We have a couple of options available, but are still trying to get a baseline, so we know if any tweaks improve the situation or not.
Is it every time I view a page that's causing it?

Yeah, its all you rather fault, gayboy

In layman terms:
Its the hourly backups
For those that understand, to get a point-in-time backup of the database (and it has to be point-in-time to maintain consistency), it is necessary to lock the database tables before the backup starts, and release the locks when its complete. In testing, this was taking less than 30s, which we thought we would get away with. Now traffic has ramped up, and an unexpected high number of users still browsing the old YaBB site, the server is working much, much harder, and its taking a few minutes. People then get fed up with waiting, so constantly hit refresh, giving the server yet more requests to deal with.