We consider conditional exact tests of factor effects in designed experiments for discrete response variables. Similarly to the analysis of contingency tables, a Markov chain Monte Carlo method can be used for performing exact tests, when large-sample approximations are poor and the enumeration of the conditional sample space is infeasible. For designed experiments with a single observation for each run, we formulate log-linear or logistic models and consider a connected Markov chain over an appropriate sample space. In particular, we investigate fractional factorial designs with $2^{p-q}$ runs, noting correspondences to the models for $2^{p-q}$ contingency tables.