Why are my dwarves not gathering wood?

The hint that allows us to answer this question is that your dwarfs are specifically ignoring wood, rather than some other resource. You should check the orders menu. One of the orders available determines whether dwarfs bother stockpiling wood at all, so make sure that's set to "yes, they should." There may also be a setting in the orders-Forbid orders submenu causing everything produced outside to be automatically forbidden. I can't remember if there is such a setting (and I can't find it on the wiki), but if there is, it would affect wood and goblinite more than anything else.

A second cause has to do with individual labors. Wood has its own hauling labor category. All dwarfs start by default with all hauling labors turned on, but it's possible that in your zeal to keep dwarfs on task rather than pointlessly moving things around the fortress, you've inadvertently turned wood hauling off on all of your dwarfs. You can check this by viewing a dwarf, and going to preferences-labor, opening the Hauling submenu (with numpad +/-/Enter), and checking what is turned on or off. If you're using Dwarf Therapist (which you should!), the hauling labors are all pretty far off to the right of each row; make sure at least someone has most of them activated. Developed fortresses tend to require a lot of hauling, so it might be a good idea to set some of those useless unskilled Peasant immigrants to the "Mover of Things" job.

Another possible cause has to do with burrows. Wood, unlike most other resources, is usually found outside. If most of the dwarfs are not allowed to go outside, then they won't be able to collect the wood. The few dwarfs who are allowed to go outside (the woodcutters themselves) will eventually get around to it, but probably not until they run out of trees they've been told to murder. But if you're new to the game, you probably haven't been mucking around with burrows yet.

The most likely problem is that they're just too busy. Cutting down the tree and hauling the wood aren't associated, so they may decide to take another task in between. They'll get to it eventually.

This worked for me (kind of). Mark all of the logs to be dumped. Once they've been placed in a dump reclaim them. The dwarves (some of them) will go start collecting them (most of the logs).

I had that problem too, and then I realized I simply didn't have a wood stockpile. If you have wood gathering enabled, they won't gather unless they have somewhere to put it. So you have to go and create a wood stockpile, and they will gather it and put the wood in the stockpile.

Thing that worked for me was simply to remove wood storage area and designate new one. 20+ dwarves rushed then immediately to pick up the wood ;-)

I noticed 2 things on my own game regarding this:

One of the things I noticed is that I had a burrow selected right over that area, and at one point, I had stationed all of my current dwarves in there. Since I got Migrants, it seems only the migrants that had the wood hauling selected gathered wood. So I had to delete the Burrow and the stockpile, and put a new stockpile down (I'll deal with the burrow later), and all of a sudden, all of my dwarves that I had wood hauling selected started gathering wood again.

The Second thing, I only noticed after it happened again WITHOUT the burrow affecting it. I put 3 Wheelbarrows assigned to that stockpile, and I had ALL of my dwarves with both the wood hauling AND the vehicle push/haul on. Then when they were done bringing what they had in their hands at the time, they stopped. I removed the wheelbarrows from it, and they all went back to it. I did this a few times to test it out, and that seemed to be the problem. (Ver.0.44.12)

If you've accidentally assigned a stockpile to have a bin or barrel amount, the dwarves will not put wood in it. This is because they can't cram a log into a barrel.