What periods of the fossil record are most lacking in specimens?
The biggest temporal gap would be (IMO) the Precambrian, specifically pre-Ediacarian. Accordingly the biggest gap in the evolutionary history is the origin of eukaryots, both because of the paucity of pre-Ediacarian formations but also because few of the early protists were likely to be fossilizable.  

From the probable apparition of life ca. 3.8-3.5 Ga (e. g. 1 for a review of Archean fossils) to the Ediacarian (ca. 600 Ma) in which metazoans (i. e. "animals") are already diverse (e. g. 2), this is more than 3 Ga of life history for which we have very little fossils (though our knowledge of this fossil record increases currently thanks, in part, to our better understanding of the "chemical" fossil record).

1: J. W. Schopf, 2006. Fossil evidence of Archaean life. Philosophical transactions of the Royal Society, B., 361: 869-885.
2: S.Conway Morris, 1993. The fossil record and the early evolution of the Metazoa. Nature, 361: 219-225.