Thank you very much for your reply, I'll definitely explore the "iotop" tool.
At the moment, however, I think that I have found the source of the problem.
One of my client's sites was an old SMF forum. It was a neglected forum, probably hacked, and so a target for malicious scripts. I explored the logs and found out that these scripts posted dozens of messages per second.
I disabled this forum and - at least for now - the server operation is normal. I hope that this was indeed the cause of the problem.