If you have imported course announcements, make sure you change their release dates. If you don't, your students will see all announcements for the whole course. From your course homepage:

Click on the caret in the Announcements widget and click on "Go To Announcements Tool"

Click on the caret next to an Announcement title and then click "Edit:"

Finally, scroll down past the Announcement content to "Availability" and edit the Start Date (the date you want the announcement to post for students). You can also edit the End Date here (the date you want the announcement to disappear for students).