Question: Does the text need to be directly on top of the line? If you open the Geometry inspector, you can change the offset so the text isn't sitting directly over the line, which would make it more legible without requiring complicated shadow effects.
The other alternative would be to make sure your text labels have an opaque (rather than transparent) background color.
To do this, select the text in your line label and select
Format -> Fonts -> Show Fonts
depending on what version of the OS you're running, the panel that appears may look a little different from what's shown here:
but the one with the help tag is the control you're looking for.
Double-click it, and the color panel will appear.
You can use the slider at the bottom to make the opacity 100%, which means the line beneath the text will be hidden.
That help at all?