Communicated by R. McKenzie We present a new solution of the word problem of free algebras in varieties generated by iterated semidirect products of semilattices. As a consequence, we provide asymptotical bounds for free spectra of these varieties. In particular, each finite R-trivial (and, dually, 1250063-1 G. Horváth et al. each finite L-trivial) semigroup has a free spectrum whose logarithm is bounded above by a polynomial function.