Ahh, a tree question.
Herodotus has it right - selecting where and what to cut is critical. Also, unless you get very lucky, kiln drying is necessary to get check-free pieces of the correct moisture content that are sufficiently large to use. Large temp variations and the variable (and, in Va , high) humidity in a barn will not simulate a kiln. High temps alone are not enuf - moisture control is key. Some species air-dry ok if stacked properly, but it has to be done carefully and air-drying is usually just the 1st step before kiln drying for hardwoods anyway.
Drying issues aside, walnut and maple are your best bets. They're chosen for stocks because they're either what we tree people refer to as 1) "diffuse porous" and don't have large vessels in the wood formed early in the year (maple) or 2) if they're ring porous the vessel density is very low (walnut). Vessels are big cells in wood that conduct water, and both hickory and oak are very ring porous with high vessel densities and large diameter vessels (which is why the rings are so distinct). Among other things, this makes them difficult to machine very smooth (unless the trees are growing v. slowly, which makes the rings narrow and the surface easier to work smooth). And, it makes working countours and abrupt changes in contour (such as found in a stock) difficult to do and still get a smooth surface. Both oak and hickory do not "retain shape" as well as the other species either, which is bad in gunstocks.
The oak is best plank-sawn and used for millwork or peeled for veneer, but you have to cut the tree with those things in mind ahead of time. Hickory makes good tool handles and cabinetry (and skis in the old days), but probably won't make much of a gunstock.