OK - so you are talking about the ergodicity of a Markov chain with respect to a finite stationary measure. One general result you should be aware of is that in this situation ergodicity of the time shift in the path space (this is essentially the definition you use - you just refer to the corresponding ergodic theorem) is equivalent to "irreducibility" (absence of non-trivial invariant subsets) of the state space, see http://mathoverflow.net/questions/74503/different-uses-of-the-word-ergodic/74503

However it does not help much. The point is that the question you ask is essentially the same as asking about a "general method" to decide whether a given probability measure preserving transformation is ergodic. For, any such transformation can be considered as a "deterministic" Markov chain (all transition probabilities are delta measures). On the other hand, as I have just mentioned, Markov chain ergodicity is equivalent to ergodicity of the deterministic time shift on its path space.

Now, there are no general results in ergodic theory which, to quote your question "can help to easily establish ergodicity". Indeed, any textbook in ergodic theory opens with the standard set of examples (irrational rotation, Bernoulli shift, Gauss transformation, hyperbolic toral endomorphism), ergodicity of each of which is established in its own way. Unfortunately, these approaches are very far from being universal, and ergodicity of certain well-known transformations is a very hard problem (the most famous example being, arguably, Boltzmann's ergodic hypothesis).