It all depends what "client" you're referring to - a pool miner or the vanilla Bitcoin client. When you mine in a pool you don't earn any of the transaction fees because the way they're all setup the owners keep that as part of their compensation. When you run the vanilla client or solo mine, then chances are (if you have port forwarding in place) you're also acting as a node.
I don't have a full grasp of how transaction fees are earned, but this is from the wiki:
Because nodes have no obligation to include transactions in the blocks they generate, Bitcoin senders may voluntarily pay a transaction fee. Doing so will speed up the transaction and provide incentive for users to run nodes, especially as the difficulty of generating bitcoins increases or the reward per block amount decreases over time. Nodes collect the transaction fees associated with all transactions included in their candidate block.
I've been running a pool miner and a Bitcoin client separately. My Bitcoin client isn't generating any blocks, it's simply acting as a p2p node. Here's a description of how a node plays a role in transactions:
Transfers are facilitated directly without the use of a financial processor between nodes. This type of transaction weakens consumer protection by making chargebacks impossible. The Bitcoin client broadcasts the transaction to surrounding nodes who propagate the payment across the network. Corrupted or invalid transactions are rejected by honest clients. Transactions are mostly free, however a fee may be paid to other nodes to prioritize transaction processing.
I wasn't sure what to expect doing this, but after about a week I had "Generated" .12 BTC. So I can only surmise that I earned transaction fees that propagated through my node?