Gmail: Open the conversation and then click the More drop-down menu. Choose Mute from the list and the conversation is archived. Any new messages sent to the conversation thread will not appear in your inbox but will be added to the archive. (Note: The conversation will reappear in your Inbox if someone replies to you only.)
Outlook: Select the thread and press ctrl + delete. The thread will be deleted and any further messages determined to be part of the same thread will be immediately deleted.