WordToWorship.com displays lectionaries which many churches follow for scripture readings.
To search for songs related to the readings for a week, first click on the "Lectionary" tab on any page and choose the lectionary your church uses. Next you can use the "Skip to this week's readings" link to jump to the current week. (You can then bookmark that page for quicker access.)
For each week the following are displayed: the service name (such as "Second Sunday of Easter"), scripture readings, and occasionally some related themes. Under the readings a link to BibleGateway can be used to read the verses. If you are logged in, a link to the verse tagging page is provided.
On the right of the table are links to search for songs related to that week's readings (and themes). Simply click on it and wait for the results to be displayed.