A population's effective size is an important quantity for conservation and management. The effective size may be estimated from the change of allele frequencies observed in temporally spaced genetic samples taken from the population. Though moment-based estimators exist, recently Williamson and Slatkin demonstrated the advantages of a maximum-likelihood approach that they applied to data on diallelic genetic markers. Their computational methods, however, do not extend to data on multiallelic markers, because in such cases exact evaluation of the likelihood is impossible, requiring an intractable sum over latent variables. We present a Monte Carlo approach to compute the likelihood with data on multiallelic markers. So as to be computationally efficient, our approach relies on an importance-sampling distribution constructed by a forward-backward method. We describe the Monte Carlo formulation and the importance-sampling function and then demonstrate their use on both simulated and real datasets.