When releasing microdata for research purposes, one needs to preserve the privacy of respondents while maximizing data utility. An approach that has been studied extensively in recent years is to use anonymization techniques such as generalization and suppression to ensure that the released data table satisfies the k-anonymity property. A major thread of research in this area aims at developing more flexible generalization schemes and more efficient searching algorithms to find better anonymizations (i.e., those that have less information loss). This paper presents three new generalization schemes that are more flexible than existing schemes. This flexibility can lead to better anonymizations. We present a taxonomy of generalization schemes and discuss their relationship. We present enumeration algorithms and pruning techniques for finding optimal generalizations in the new schemes. Through experiments on real census data, we show that more-flexible generalization schemes produce higher-quality anonymizations and the bottom-up works better for small k values and small number of quasi-identifier attributes than the top-down approach.