A "theological position" would be an opinion of some sort (that's the "position" part) about something (say, existence or lack thereof) specifically about God (that's the "theological" part). "Either God exists, or God does not exist" is a theological position, because it contains the premise that both sides of the disjunction make sense. Put in metaphysical terms: that it's possible that God exists, and possible that God does not. But I'm not sure the agnostic is entitled to that much.
This is because the agnostic looks to me to be committed to the view that "Knowledge about God's existence is impossible." I think this is necessarily true about the agnostic because it makes no sense to say, "I choose not to have a belief one way or the other about X." As Socrates insists, one believes what one believes, whether one wants to or not. The attempt to reflect on our own beliefs, to honestly and courageously evaluate our reasons for holding them, is the beginning of philosophy. That is why "Because my family raised me to believe in God" is not an adequate answer to the question "Do you believe in God?" The question is about one's beliefs themselves, not the etiology of those beliefs, although that may be revealing (as it is, embarrassingly, in the example).
If this is so then another problem for agnosticism is that it is a consequence of a general epistemological attitude, that is, an attitude towards knowledge in general, and nothing particular to do with God. Aristotle's objections to Plato's metaphysics, Hume's objections to 17th century rationalism's metaphysics, are epistemological arguments with general application. Aristotle and Hume, hearty philosophers both, breathed deep and followed Socrates' dictum: they concluded (for closely related but interestingly different reasons) that they believed that various putative entities did not exist. They were willing to accept the consequences of the epistemological standards that they had set for themselves.
The agnostic wants to be a kind of sceptic: not sceptical of God's existence, but sceptical about the possibility of knowledge of God's existence. The move is to avoid the unpleasantness of denying God's existence by denying the possibility of knowledge of God's existence. Wittgenstein would say, "When you say that asserting God's existence or denying God's existence is impossible, because there is no way of knowing which possibility is fact, you are (merely) stating that it makes no difference, that neither proposition carries any meaning because there are no pragmatic consequences either way." That is he would apply his general criticism of sceptical arguments. In fact Wittgenstein holds that propositions about spirituality are impossible for the same reasons that he holds that propositions about aesthetics, ethics and phenomenal experience, for examples, are impossible. But a crucial point here is that he denies that this makes them insignificant (as Hume or A. J. Ayer, say, might do): he affirms the great significance of many aspects of experience that lie beyond the bounds of language.
Where does this leave the agnostic? (I am fighting off the urge to go on to Kierkegaard.) The agnostic cannot say, "I believe that God might exist or God might not, but I believe that knowledge of which is true is impossible." This is self-contradictory. In order to (really) believe that God might or might not exist, one must believe that there are (somewhere, somehow) reasons for believing one or the other. But the agnostic must claim that there are no such reasons, else why not examine them with Socrates and the gang? (Just as an aside, I think that there are reasons for and against believing in God: thus I am not agnostic, even if I have not reached a conclusion.) No, the agnostic is simply refusing to examine his or her own beliefs. Pascal was right: just doesn't want to get into trouble. Agnosticism is a refusal to do theology, not a theological position.