Let's see if this works:
The gods are A, B and C. A(T) means A is the one who always tells the truth, A(L) means A always lies, A(R) means A answers randomly true or false.

Start with some true assertion, such as "Water is wet".

Ask this question of B: "If I asked A if water is wet, would A answer 'yes'?"

Here are the options:

1. B(T) and answers 'yes'. Then A must be A(T) therefore C(R)
2. B(T) and answers 'no'. Then A must be A(L) therefore C(R)
3. B(L) and answers 'yes'. This would mean A(L) also so could not happen (**)
4. B(L) and answers 'no'. Then A is either A(T) or A(R)
5. B(R) but this time answers truthfully and answers 'yes'. Then A must be A(T) therefore C(L)
6. B(R) but this time answers truthfully and answers 'no'. Then A must be A(L) therefore C(T)
7. B(R) but this time lies and answers 'yes'. Then A must be A(L) therefore C(T)
8. B(R) but this time lies and answers 'no'. Then A is A(T) therefore C(L)

** Oops, spotted at least one mistake. The assertions about A missed out the complication of A(R). Based on the assumption that A is not A(R), then the rest holds and only statement 4 doesn't identify all cases. Thus the second question just needs to establish whether A is answering randomly and also resolve A's identify for question 4. So not quite there yet...

__________________
Spike
:)