Sequential sensing algorithms are developed for OFDM-based hierarchical cognitive radio (CR) systems. Secondary users sense multiple subbands simultaneously for possible spectrum availabilities under hard misdetection constraints to prevent interference to the primary users. Accounting for the fact that the sensing time overhead can often be significant, a novel performance metric is introduced based on the effective achievable data rate. An optimization problem is formulated in the framework of optimal stopping problems to maximize the average effective data rate by determining the best time to stop taking samples and proceed to data transmission. A basis expansion-based suboptimal algorithm is developed to reduce the prohibitive complexity of the optimal solution. The numerical results presented verify the efficacy of the proposed approach.