does a $100 bill hold any "personally identifiable information" barring some trace DNA or fingerprints?
Every bill has a unique identifier on it. Every time you withdraw from an ATM, those IDs are associated with your real name. Every time you purchase something, the retailer deposits those bills right back in a bank. Occasionally, bills may be passed around privately before landing in a retailer's hands, but this actually enables data miners to determine with whom you financially associate. Cash is no panacea of privacy.
Imagine a graph in which accounts are the nodes and transactions are the edges. Cash tells the Feds who owns every node. BitCoins tell the Feds (and everyone else) about every edge. The latter information quickly loses its value when nodes are popping up willy nilly with no real names attached to them, and faster than anyone can pin names to. You just have to programmatically keep your BitCoins at a recent node, ahead of the wave of nodes that have been identified.