Is stuff like this even necessary?
Probably not. But as a thought experiment, it could be quite entertaining.
Most obvious choice would be to use snail-mail right? But that's just boring, I guess OP wants something more extraordinary.
So how about music? I mean rhythms, big-ass bells, gongs or war drums?
Should do the trick, problem is that any bystander can hear this, using some sort of cipher would be required.
There is also pigeon mail...

ridiculous but may work quite fast, although eagles or hawks may hunt your courier down

Nevertheless electricity will be mandatory at some point, no escape. Besides when we loose the electrical grid worldwide, bitcoins will be the last thing we would worry about.
And as my predecessor NeuroticFish said, there is also an issue with trust.