I don't see why the linear effect shouldn't be there. After all, you can envision the rotational effect as a following up of infinite linear effects. At any instant two bodies in circular motion wrt each other, say the Earth and the rest of the universe, are having a linear momentum wrt each other.  
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.  
Notice that the cause of the effect depends on the frame of reference. If you find yourself in at rest wrt to the stars around the two plates it are the plates causing the effect. If you find yourself at rest wrt the plates, it are the moving stars that cause the effect.
Likewise, when you find yourself on Earth (i.e., the frame co-rotating with the Earth), it's the rotation of the rest of the universe which causes the effect (or the sequence of linear motions in slightly different directions). Though the rest of the universe can't really be in a circular motion around the Earth, obviously.   
Or look at the spinning bucket with water inside. The water inside has risen. You can say that the cause of the water rising is the fact that you have set the bucket in motion. But you can just as well say (when you are happily spinning along inside the bucket) that it's the motion of the rest of the universe that causes the water to rise. If you **really** were to set the rest of the universe in circular motion (which is very difficult I guess but for the sake of thought) the water in the bucket would rise too. So when your inside a spinning bucket with water, it's safe to say that the cause of the effect is that someone has set it in motion. Likewise, when you find yourself in the frame co-rotating with the Earth, it's safe to say that it's the Earth rotation that causes the frame dragging. But it can be described in two ways.