How big a problem is wood embedded in a basement concrete floor?
The concrete floor shouldn't be effected by the embedded lumber even if it should deteriorate. If the concrete floor was poured to an appropriate thickness and mixed correctly it's best to leave the wood in place. From your photo the first thought I had was they were used as striking boards when the floor was poured, but most people remove them after the concrete is leveled. On the positive side; they can be used as attachment points if you ever decide to install sleeper's for a finished floor.