A recent article in PloS Computational Biology suggests that memory is encoded in the microtubules. "Signaling and encoding in MTs and other cytoskeletal structures offer rapid, robust solid-state information processing which may reflect a general code for MT-based memory and information processing within neurons and other eukaryotic cells."
They argue that synaptic connections are transient compared with the lifetime of memories, and therefore memories cannot be stored in them, but in some more persistent structure. The structure they suggest is the phosphorylation state of sites on microtubule lattices within neurons. And that's about as much of the technical detail as I feel able to summarise. It's not all speculation, they report technical work on the structures of these cellular components. Total memory capacity would be somewhere upwards of 10^20 bits (or in more everyday units, 10 million terabytes), depending on the encoding, of which they suggest several schemes.
Journalistic writeup here.
Note that Stuart Hameroff, one of the authors, is known for his proposals for microtubules as the mechanism of consciousness through quantum effects (and with Penrose, quantum gravitational effects). The present paper, however, is solely about memory and does not touch on quantum coherence or consciousness.