Does that mean it's illegal, or that they'll just come knocking to investigate?

I wonder if the "it's my money, I can withdraw it if I want" argument is good enough to send them on their way? (in addition to $1,000 being such a small amount as to be less-than-trivial when it comes to the overall problem of money laundering).

I think the limit is 10k and withdrawals over 1k get bundled. If you hit the limit you get reported to the Unità di Informazione Finanziaria (Financial intelligence unit) and what they do is under their discretion.