We consider a spin system on a lattice with finite-range, possibly unbounded random interactions. We show that for such systems the Glauber dynamics cannot decay to equilibrium exponentially fast in L_2 even at high temperatures. Additionally, for one-dimensional systems with unbounded random couplings we prove that with probability one the corresponding Glauber dynamics has a fast (subexponential) decay to equilibrium in the uniform norm, provided that the distribution of random couplings satisfies some exponential bound.