Deleting a recent announcement post in Google Sites is really easy to do, but isn't the easiest thing to find. A post in the Recent Announcements is an actual page. Every time you create a post, you create a unique page. So all you have to do is navigate to that specific post and then delete it. Below is a tutorial on deleting your posts.
1. Make sure you are logged in as a site administrator. Navigate to the page that has your recent posts on it. Click on the "Recent Post." In this case I would click on the "fridayearlyrelease" link. This would then take me to that specific post.
2. Click the "More" button at the top left of the page.
3. Now just choose the "Delete page" option from the "drop down" and you page and the recent announcement will be removed.