The hightly customizable plugin xili-postinpost is able since version 0.9.2 to display the result of the query according time or date.
Below one attachment appear in the morning and another appear in the afternoon
this is afternoon
The date kept is the local date of the server that is set in General Settings of WP dashboard. Two shortcodes are present in content of the post. As here, xili-pluginpost is compatible with WP 3.0 multisite mode.