Log into your Blackboard course and go to the Course News page.  Locate the box labeled My Announcements and click the more announcements link in the bottom right corner of the box.

You will now see a screen labeled Announcements. To delete an announcement, locate the one you wish to delete and hover over it.  Click on the chevron button that appears to the right of the announcement name and select Delete from the menu.  You will then be asked to confirm the deletion.