After posting this, I've managed to become thane in multiple holds without joining the war on either side, but I could not become thane of Eastmarch without taking a side, for I have already completed Blood on the Ice, and so my original advice still stands.
I've already done the neutral route: basically, you need to avoid picking a side and get through the main questline (with High Hrothgar). Eventually, you'll have to moderate a meeting between the jarls at High Hrothgar, which effectively ends the civil war questline without joining a side. After that, you can go to the jarl and he'll give you the offer to become a thane.
Do the main quests, once you get to the mission where you ask the jarl of whiterun if you can use the reach to capture a dragon he'll tell you he can't worry about that while the war is still going on. That's when you go to the Grey beards and ask them if you can hold a council to end the war and basically make a peace treaty. Then you can go to windhelm and do as needed to become the thane and purchase property in the kingdom. You're welcome.