Bitcoin is not an anonymous digital currency. It is a pseudonymous digital currency that is traceable.
With proper precautions, it can be used more or less anonymously.
Exchanges add an additional weak spot when trying to use the currency anonymously.
It can be done but it is expensive and time consuming -- increasingly more so when the amount to convert and the level of privacy required increases.
Subject A then converts payment (USD) to BTC
For A to obtain BTC anonymously there are scenarios:
- In the U.S., Canada and Australia there are cash deposit methods at banks or post offices where no ID is required. Those funds can then be used to purchase bitcoins from an exchange. Not all exchanges allow purchases when using Tor to access the site anonymously, so know that. Just because an ID isn't required doesn't mean you would never be identified (banks have cameras, for instance).
- Cash in the mail. There are exchanges and individuals who will accept cash by mail and will send bitcoins to the bitcoin address included in the envelope. Get-Bitcoin (USD), BitcoinNordic.com (EUR) are two such examples. Individuals will accommodate this as well. There are risks, of course, in that the post office fails to deliver or that the recipient isn't honest.
- Person-to-person trade. There are many trusted individuals on Bitcoin-otc or other services where a trade, person-to-person, can be made. Making use of a dead drop will help to protect your anonymity if required.
To further protect the privacy of the transaction, the received coins can then be mixed with others. There are numerous ways this can be achieved, including using a mixing service or employing an EWallet. Using EWallet for this specifically are addressed here:
Subject A sends BTC to Subject B's BTC Wallet Address
Same as above. Use Bitcoin with Tor to essentially eliminate any chance your IP will be associated as the node your transaction was "first relayed by".
Subject B converts BTC to USD (Cash)
This can be the most difficult step if a high level of anonymity is desired. Certainly, the received coins can be further mixed like has been explained above. But there still would be a point where cash is physically retrieved in exchange for bitcoins.
One method, just like above -- is to use #bitcoin-otc to find an opportunity to do a person-to-person trade. Green Dot Moneypak codes can be purchased from BTCPak.com anonymously, but from there they can't be used with reloadable cards without first registering the destination card.
There are cash out methods where you would receive cash in the mail, but the mailing address you provide is where this method no longer remains one that preserves anonymity. If a physical cash exchange isn't required then there are numerous goods and services available with payment using those bitcoins being a possible alternative.
Here's a related artcle:
(Notice how nearly every service listed in that article is now defunct? Anonymity is a double edged knife. Some "mixers" have been scams and some exchanges never delivered. And because they too operate anonymously they can easily scam.)