How can I empty the trash without first opening it?

The only way I've seen how to empty the trash is to use the Empty Trash button that is displayed after choosing Go | Trash in Nautilus. If there are too many items in the trash it can take a long time for the list of files to load, especially when Nautilus defaults to generating thumbnails.

Is there a way to empty the trash that doesn't require opening it first?

If you enable a trash icon on the desktop, You can right click on it and select 'Empty Deleted Items'. Also, within the file manager, you don't have to navigate into the Deleted Items folder; you can right click on the Deleted Items place on the left hand side and select 'Empty Deleted Items'.

If you have the "Trash Applet" on the lower right of the screen by default, you can right-click and get a menu with "Empty Trash". In the shell you can just

$ empty-trash

In Unity, a trash icon is provided at the bottom of the launcher. Right-clicking this will open a quicklist in which the "Empty Trash..." option is provided.