It's possible to give someone your private keys either by sending them a wallet file or by writing them down. The problem is that unless you do a transaction through the network double spends are possible. This means writing down the info essentially makes it a check not cash, because you have to trust that the funds will still be there when you try to use it. You can't pass these notes on over and over since you would have to trust everyone who had seen it. So really it's even less secure than a check.

I don't mean there isn't potentially some solution, only that the obvious way to do it results in insecure checks, not cash.

Oh, and don't do anything with regular dollars. It's weird, confusing and not helpful to your goal here.