Is there an equivalent to derandomization for quantum algorithms?
There are certain classes of quantum gates which can be simulated efficiently with a classical computer. If no entanglement is present, a computation with pure states (i.e. not random states) can be simulated efficiently. Classical gates reversible gates are a subset of quantum gates, and so can obviously be simulated efficiently. These two examples are pretty trivial, however there are a number of non-trivial gate sets known.


Valiant gates as mentioned in Joshua's answer
Clifford group gates (see arXiv:quant-ph/0406196)
Match gates (see arXiv:0804.4050)
Commuting gates, etc.


Basically, most sets of operators which generate only some small subspace of $SU(2^N)$ tend to be simulable, while any that generate $SU(2^N)$ are as hard as general quantum simulation of N qubits.

It seems very unlikely that quantum mechanics is efficiently simulable, and so such a dequantization program will likely be impossible in general. There is however a regime where this has worked, which is with interactive proofs. Several different kinds of interactive proof systems with quantum verifiers have been shown to have the same power if the quantum verifier is replaced with a purely classical verifier. For an example of this, see Jain, Ji, Upadhyay and Watrous's proof that QIP=PSPACE (arXiv:0907.4737).