DIY speakers of solid/reclaimed wood?

That is a small speaker and the wood is thick enough with enough structure that warping won't be a problem.  A cabinet that small with front and backs attached will be fairly inert. I've built tons of speakers and even though I really don't use solid woods.  I could, but would properly brace everything to keep down issues that arise with solid woods. 

There are a few technics to prevent expansion cracks. The most probable  easiest is to rip the wood into stips,  maybe 4 to 5 inches wide and regluing them to fairly well match wood grain.  These small strips do not expand like something that is say 12 inches wide.  
There are other things that you can do,  it just takes a bit of know how.