BTCDice works using the UTC timestamp from transaction to determine day of transaction (in UTC) and then process using secret of that day (in UTC).
Transactions don't have timestamps though. You client may make a note of the time that it first saw a particular transaction, but my client may put a different timestamp on it. I don't think there's any way we can safely agree on the exact timestamp of a transaction, since it's not part of the transaction itself. All we can agree upon is the timestamp of the block that includes it.