Neil - Is there something in say the manual for exorcists/racoulta about this?  Do you have any objection to relief carving if those requirements are met?  And, do you think the devil will respect a relief carved crucifix if those requirements are met?  I think relief carved crucifixes are used among the eastern catholics as well, not just schismatic orthodox.  I prefer the relief carving for the reasons I mentioned.  I would rather have a hand carved affordable item than an expensive modern wood crucifix made by a computerized carver.  And, I would certainly rather have a relief carved crucifix over a resin crucifix.  I think it is the way to go.  Carving a modern crucifix is extremely difficult.  These are doable.  

   

  

.
Regarding choice of wood species, I would recommend a strong hardwood for the cross, such as oak or American black walnut, ash, or even bamboo (might have to be laminated strips). If you want a very strong and hard wood that carves pretty well, maple is a great choice and pretty available. Be sure your blades are sharp, because maple will burn quickly on a table saw with a dull blade. When used for flooring, maple is more durable than oak because it's tougher.
.
A popular wood used traditionally is ebony (black in color) but that's pretty hard to come by these days. For lighter color wood a comparable species is boxwood but that's pretty scarce too.
.
Olive wood is widely used in crucifixes made in the Holy Land but I don't know where you can find that for sale in the States. If someone is cutting down an olive tree you could get some of that and go from there but if you're in a hurry seasoning wood can take a LOT longer than you want to wait. (Like years) Compare olive wood to a bit less strong than oak, maybe like beech (common in Europe) or mahogany. FYI in the old days the best slide rules were made of mahogany core or bamboo core, with ivory-like overlay (made of plastic that looks like ivory). Mahogany and bamboo were the best because of their inherent self-lubricating property, making them good for the sliding action of a slide rule.
.
If you want to carve your own corpuses you're in for a huge challenge, it seems to me. And if you do that with wood the product can be very fragile. That's where I would say maple might be your best choice, offhand.
.
One trick I found that's pretty neat is to buy the crucifix ready made from a Catholic supply shop, and then you can custom mount it onto a river-run stone for a base. Just drill a hole with a masonry bit and bore a countersink with a larger carbide-tip bit, then insert a long drywall screw through the stone into the base of the crucifix. Glue some rubber or felt feet on the bottom and you're done. That gets you off the hook without having to make the whole crucifix.
.