That would be possible with assets issued on the blockchain by a gateway service, for example if a bank takes USD deposits and sends you USD colored coins in return. You can then trade those USD coins just as you would any other coin on the blockchain, safely and transparently.

Of course, this means trusting the service to honor the colored coins and not to default if you want to redeem them for "real" USD. However, this is still much safer than other centralized financial institutions since the accounting is all done publicly on the blockchain, and everyone can tell if the service is ripping you off or not. And there could be a diversity of gateway providers to use, for instance commercial banks or even the Federal Reserve.

There is no way to get away from some amount of centralized trust when using fiat, we can't magically turn it into a cryptocurrency. However, USD-backed assets on the blockchain is the closest we can get.