Reading your question again, and looking at the comments, I think your question is slightly misinformed.
The standard bitcoin QT client is a node, which means that it receives the inventory of all transactions and relays all valid transactions it requests to receive (from the inventories). That's a bit of a mouthfull but read it carefully.
This has nothing to do with your wallet. It is a standard function of the full node on the peer to peer network.
What you appear to be mixing is the functionality of the wallet and the functionality of the node. These are two different things.
Bitcoin QT is unlike other (lightweight or SPV) clients which only request information specific to the addresses they hold, when trying to syncronise themselves with the network.
If you are looking to essentially print all transactions that are being relayed through your node, I think this would be pretty straitforward using something like Node.js or just running a continuous regular process which queries Bitcoind using RPC.
What do you mean by "stream"? What do you want to do with those transactions? What do you want to be done for you? – Nate Eldredge – 2014-04-02T15:04:04.880
@NateEldredge I want to do something similar to how blockchain.info displays transactions in real time. I just didn't know if the client exposed this anywhere, or not. I can see them in bitcoind's console - do I monitor console output? <shudder/> – Avram – 2014-04-03T00:36:40.927