Dead, fallen wood can be good firewood. It depends on whether it's wet or dry. Wet or dry depends, not on rainfall, but on contact with the ground. It also depends on how long the log has been lying on the ground. The longer a log has lain on the wet ground, the more waterlogged and rotten it will be.
A log lying on the ground will likely be too wet for good firewood. If it has fallen recently enough that the bark is still well-attached, it may be be possible to dry it out and use it. The easiest way to dry out a log is to prop up one or both ends off the ground, in situ. If you prop up one end, of course the end still on the ground will stay wet, and not make good firewood. Use a convenient rock, or another log (keeping in mind that if the other log is rotting, the newer log will likely not dry very well at the point where it makes contact with the rotting log), or the crotch of a nearby tree (only if the tree is large and sturdy enough to support the weight of the log without damage).
A log with one end propped up will be relatively dry at that end, and relatively wet at the end on the ground. Logs often end up partially propped off the ground, either because one end is still attached at the stump, or they fall across other trees or logs, or because some of the attached branches support the log off the ground. Any branches on the ground will be not good for firewood, but branches still in the air can be good for firewood.
As you get more practice at collecting wood for fires, you will get a better sense of how dry wood needs to be to make good firewood. As you develop this sense, I recommend having your fires in an outdoor firepit, rather than an indoor fireplace. You could do an outdoor test burn of each "batch" of firewood to make sure it's suitable for indoor use. Mediocre firewood is more prone to smoking (which may cause chimney build-up, which eventually causes chimney fires), and popping (which can send sparks out of the fireplace). Keep in mind that conifer trees have pitch in them, which makes them burn easily, but causes lots of popping and flying sparks. If you have conifer wood, save it for outdoor fires.