How does the Bitcoin client make sure transactions were accepted by the network?
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.