Which organisms have the neuroanatomy Roger Penrose supposes play a role in consciousness?

Microtubules are a structure in the cytoskeleton, they are rope like polymers that grow to a length of about 25 micrometers (25000 nm), and have an outer-diameter of around 25 nm. For comparison, the mean spacing between atoms is on the order of 0.1 to 0.2 nm; so the micro tubule really is micro: about 200 atoms across. In terms of quantum effects though, this is pretty big but not unreasonable. Researchers commonly use quantum dots to play with quantum effects, and these are typically spheres on the order of 10 to 50 atoms in diameter. Note, that we don't know how to couple 5000 quantum dots in one coherent chain (how many you would need to get the length of a microtubule).

So microtubules are small, but they are common! Microtubules are found in all dividing eukaryotic cells and in most differentiated cell types. In other words, that mosquito you just smacked and that philosophers always give as an example of something non-conscious is full of microtubules. This should raise some red flags, but we don't need to go into more detail of microtubules to discredit Penrose and Hammeroff. However, if you love cell biology, take a look at Desai & Mitchison (1997).

So microtubules are probably a bad basis, but why did Penrose want quantum effects in the brain? In The Emperor's New Mind, Penrose suggests consciousness is non-algorithmic and suggests that a magical quantum computer could do these non-algorithmic tasks. The reason I use 'magical' is because a real quantum computer is Turing-complete, if a classical computer cannon solve a problem then neither can a quantum one (of course, if a classical computer can solve a problem, then quantum one can as well and might be able to do it qualitatively faster). For a nice computer science debunking of this part of Penrose's argument take a look a Scott Aaronson's lecture notes.

Why did Hammeroff want quantum-ness? To avoid dualism in explaining consciousness. However, he has gone so far down the reductionist rabbit-hole that he popped out on the other side. He arrived at the same 'magic' we feared in dualism except now he called it 'quantum mechanics'.

The biggest irony of this approach is that Penrose was inspired in many ways by Schrodinger's beautiful take on life. Although Schrödinger does bring in quantum mechanics (both as a useful reduction and as an analogy) he uses completely different parts of it (he uses the discritization of energy levels, and specially avoids issues of the uncertainty principle and superposition of states that made him famous). Schrödinger would completely disagree with Penrose and Hameroff::

[I]f we were organisms so sensitive that a single atom, or even a few atoms, could make a perceptible impression on our senses -- Heaves, what would life be like! To stress one point: an organism of that kind would most certainly not be capable of developing the kind of orderly thought which, after passing through a long sequence of earlier stages, ultimately results in forming, among many other ideas, the idea of an atom.

This response can be made precise through quantum decoherence (Tegmark, 2000) and there is little regard for the physical importance of quantum mechanics in the brain (Litt et al., 2006) although Hammeroff (2007) still defends it.