Notifications can be cleared in bulk for both Inbox notifications and sidebar notifications.
To clear all notifications in your personal inbox, select the Mark All as Read icon in the Unread tab of your Inbox.

To clear unread messages for a Contact Center or a Department, click the double-checkmark icon to mark all as read.

Sidebar notifications can be cleared for the Favorites and Recents sections of your left-hand sidebar.
Hover over the section title to see the double-checkmark icon
Select the double-checkmark to clear all notifications in that section