When announcing a transaction, how does the Bitcoin client make sure network peers accepted it? For example, if I put in no transaction-fee but all peers I’m connected to require a fee and therefore reject my transaction, how will I notice it?
The client doesn't ensure peers accept it. If none of your peers relay the transaction, then it will likely never get into a block and will remain unconfirmed forever. Your client will continue to make attempts to relay the transaction periodically, and you may eventually connect to a client that will relay transactions that have no fee or your coins may eventually get old enough that the standard client will relay them with no fee.