We study the Glauber dynamics at zero temperature of spins placed on the vertices of an uncorrelated network with a power law degree distribution. Application of mean-field theory yields as the main prediction that for symmetric disordered initial conditions the mean time for reaching full order is finite or diverges as a logarithm of the system size N, depending on the exponent of the degree distribution. Extensive numerical simulations contradict these results and clearly show that the mean-field assumption is not appropriate for describing this problem.