What happened is that Tertiary/Quaternary were dropped first, together, by the International Commission on Stratigraphy (ICS) formally in 2004 (see Gradstein et al. 2004a for instance, along with the 2004 edition of "A geological timescale", Gradstein et al 2004b) but then Quaternary was reinstated as a formal Period in 2009.
Geowhen database has a nice page on the pre-2009 story (I'm not sure this database has been updated since, unfortunately). To summarize what they say: Giovanni Arduino (1759) separated geologic time in Primary/Secondary/Tertiary; in 1829, Jules Desnoyers added the term Quaternary; the term has been used since then to describe the recent period of glacial/interglacial cycles. But since then, the concepts of stratigraphy evolved and came the need to be able to pinpoint the boundaries between periods/systems/stages accurately and because they were different definitions on the Tertiary and Quaternary top and bottom boundaries in the literature, the ICS opted for a completely different system: Paleozoic/Mesozoic/Cenozoic, with the Cenozoic separated into a Paleogene and a Neogene.
But in 2009, the Quaternary was reinstated (see Gibbard et al 2010): the rationale, if I understand correctly, was that the US Geological Survey and the Geological Society of America were still using the Tertiary/Quaternary System and reinstating the Quaternary was somewhat of a compromise. Since the Paleogene and early Neogene had already many GSSPs (which are precisely defined boundaries, with a golden nail planted at the precise datum in the type section), reinstating the Tertiary as well would have been counterproductive at that stage.
To add to the confusion, the "new" Quaternary, as defined from 2009 on, has his bottom boundary defined as the bottom boundary of the Gelasian (the "old" Quaternary bottom boundary corresponded roughly to the bottom of the Calabrian, thus 750kyr younger than the new one), making it overlap with part of the "old" Tertiary.
According to the 2012 edition of "A geological Timescale" (thus Gradstein et al. 2012), the rationale for the lowering of the base of the Quaternary to the base of the Gelasian is because of a now well-established cooling at 2.6/2.7Ma, which is considered to be the beginning of the first glacial/interglacial cycle.
Current ICS chart. © International Commission on Stratigraphy, 2014.