Driveway expansion joints are rotting wood: What to replace the wood?
I have been caulking expansion joints in concrete for 15 years.
This is what you do:


Get the wood out
Clean the top 3/4" of the interior walls of the joint with a wire brush 
Install "closed cell" backer rod into the joint. You want it to be about 1/8"-1/4" larger than the joint. Roll it down in from the side is easiest. The top of the rod should be about 3/4" below the top of the concrete. 
Blow the crack out with a leaf blower. (the backer rod should fit tight enough to stay in place as long as you dont blast it head on)
Use professional grade urethane or silicone caulk for concrete. The best thing is probably Dow 888 silicone but its very expensive. Urethanes are cheaper and almost as good. Sika brand Sikaflex 1a or Sonneborn brand NP1 are great
Caulk depth in the center of the joint should be 1/2 the width of the joint but no deeper than 1/2". People are tempted to think deeper is better but it will cause bubbling and premature failure. If it is too thick it will not stretch easily and therefore pull away from the concrete. 
Tooling the caulk is necessary. It will force the caulk into the pores of the concrete and it will smooth out lumpiness. Use a metal or rubber spatula or a tablespoon. 
Self levelling urethanes are available but the have the consistency of pancake batter so your joints have to be pretty level to use it. You'll also have to dam up the ends of your joints. Plus you will have to be on the lookout for rain, bugs, and blowing leaves for a couple of hours after you finish. 
2x4s are a lousy choice for expansion joint because they are so wide. You are going to use a lot of caulk.