Bitcoin makes it public record.
Regulating Bitcoin transactions is extremely easy due to the public ledger. You simply make it illegal for exchanges to transact with coins that are known to be stolen or otherwise flagged. Then all exchanges are forced to blacklist coins on that list, as well as descendant transactions from those blacklisted coins.
Now there are two classes of addresses on the blockchain: Those that are clean, and those that are tainted with illegal activity transactions. The tainted coins are worth less than untainted coins as fewer and fewer exchanges will touch them.
Ironically, Bitcoin speculators will support this move because it reduces the supply of Bitcoin. Reducing the supply of Bitcoin increases the relative value of their clean Bitcoin.