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I don't see why the linear effect shouldn't be there. After all, you can envision the rotation as a following up of infinite linear motions. At any instant, a body in circular motion is composed of bodies with linear motion. If you envision two extended massive plates moving parallel through space, for sure you will be dragged along in a direction parallel to the plates if you find yourself in between them (or outside them but in between you'll experience no "normal" gravity). You will end up with the same velocity as with which the plates are moving.
This goes to show that space is connected to matter and not to mass, which is just a property of matter. If only mass were the cause of curvature, then no framedragging was present.