If I had a chance to get some Oak I would take it no mater how green it was. Just because it is such a rarity in these parts. But I would only take a cord or so and no more. My problem is also one of space. I don't have the room to store years supplies of wood, 3 years ahead!. My wood shed can only accommodate 5 cords. I can, and do, pile a cord or two extra off to the side in my splitting area, but that is wood I either burn right away or it is some of the stuff that isn't quite seasoned and will be split and stacked once I use up some of the wood in the shed.
I prefer to cut pre-seasoned, standing dead, softwoods like pine mainly for that reason alone (no space to store for years), but also pine is so readily available and cuts and stacks well.
With out even looking I have been offered other unseasoned hard woods like apple and walnut, these "scrounge" wood are often very branchy trees, lots of crotch wood and irregular shaped pieces that are hard to cut and stack, and usually when they are offered come with the condition that you must take all of it, not just go in and take the nice rounds. In the time it takes me to cut and deal with a half a cord of this green stuff I could have a full cord of the pre-seasoned pine cut, stacked, and ready to burn sitting in my wood shed.
But I'll still take a cord or so of green oak if someone offers, just for the novelty.