We define a "threaded blockset," which is a novel generalization of the classic notion of a multiple alignment. A new computer program called TBA (for "threaded blockset aligner") builds a threaded blockset under the assumption that all matching segments occur in the same order and orientation in the given sequences; inversions and duplications are not addressed. TBA is designed to be appropriate for aligning many, but by no means all, megabase-sized regions of multiple mammalian genomes. The output of TBA can be projected onto any genome chosen as a reference, thus guaranteeing that different projections present consistent predictions of which genomic positions are orthologous. This capability is illustrated using a new visualization tool to view TBA-generated alignments of vertebrate Hox clusters from both the mammalian and fish perspectives. Experimental evaluation of alignment quality, using a program that simulates evolutionary change in genomic sequences, indicates that TBA is more accurate than earlier programs. To perform the dynamic-programming alignment step, TBA runs a stand-alone program called MULTIZ, which can be used to align highly rearranged or incompletely sequenced genomes. We describe our use of MULTIZ to produce the whole-genome multiple alignments at the Santa Cruz Genome Browser.