Sending payments to fake addresses provides a way to broadcast short encoded and optionally encrypted unalterable messages to the world anonymously. Have security codes for the organization committing genocide, but scared you'll be killed if the publisher accidentally reveals your identity?

Publish them with Bitcoin.

That is an interesting application of Bitcoins that I hadn't thought of. Sending 0.0000001 BTC to some "trigger" address to detonate some weapon or to trigger a bot/virus to send a simple (even "complex") message when the value is received. I could think of several "covert" applications to something of that nature, and it would be difficult to trace the origin of that transmission too.
Other peer-to-peer networks could work along a similar approach, but the current organization of Bitcoins assures that everybody in the entire network is aware of the transaction in a relatively short period of time (usually just an hour or less) and even temporarily split off sections of the network will eventually get the message out.
I've heard about this being done in the financial industry where somebody will send out an ask/bid order for stocks or mutual funds that is far outside of normal trading boundaries as a means to transmit a message in some fashion (aka bidding $5 for Berkshire Hathaway Class "A" stock). I'm sure other examples could be given but it is something that has already been discussed in other forums and even has fiction written about it. Tom Clancy wrote about it in terms of a trigger event for a melt-down of Wall Street in one of his novels.
It is the persistence that nodes have for transmitting transactions to each other that would be beneficial here. If anything, this is a consequence of becoming more anonymous in terms of the transactions.