Quote Originally Posted by kybarry
Asking that question has certainly helped me. On my last test, I nearly wrote "Он переехал за границей" but I realized the question is куда, which is in accusative, so I said за границу instead. By knowing which cases verbs take, it can be very helpful.
But I apparently made a mistake with this one, which my professor was a bit stumped on. "Куда ты повесил картину?" I answered: "Над стол" because I figured Куда was the question so it should be in accusative, but the answer was "Над столом." She asked someone and said it had something to do with "displacement," can anyone elaborate on this for me?
This is why the question technique will not work for non Russians. You are trying to ask wrong questions. The questions "куда", "где" and "откуда" have nothing related to the cases. The questions related to the cases are "кто" and "что" only ("what" and "who").

Он переехал за границу. - Он переехал за что? (за кого?) - Accusative case.
Он повесил картину над столом. - Он повесил картину над чем? (над кем?) - Instrumental case.
Он повесил картину около окна. - Он повесил картину около чего? (около кого?) - Genitive case.