  We study the Dirichlet problem for discrete harmonic functions in unbounded
product domains on multidimensional lattices. First we prove some versions of
the Phragm\'en-Lindel\"of theorem and use Fourier series to obtain a discrete
analog of the three-line theorem for the gradients of harmonic functions in a
strip. Then we derive estimates for the discrete harmonic measure and use
elementary spectral inequalities to obtain stability estimates for Dirichlet
problem in cylinder domains.
