If you ask about alternative methods to sample the same  "non-standard" distribution as MCMC does, then a not Markov chain based method, which may prove to be better in a specific setting, is for instance importance sampling. Specific setting here would mean that you have a good concept of a target distribution, i.e. you can come up with a somehow similar PDF.  

Let me point out that what you actually sample is the stationary distribution of the chain. In other words, you need a Markov chain whose stationary measure coincides with your target distribution. You let the chain evolve long enough (order of mixing time) for it to reach equilibrium and then you get your sample as the final end-point of the chain trajectory in the state space.

One way to construct such a chain is using the well-known Metropolis rule. This gives you a chain that is reversible with respect to the target distribution. Under irreducibility and aperiodicity, you are guaranteed that the reversible measure is the unique equilibrium distribution.

The big problem with that is the mixing time. Usually reversibility implies exponential exit from local maxima of the distribution (as a function of the state). As a consequence, it may take a very long time (exponential in the size of the state space) for the chain distribution to be really close to the target one.

Evidences are that irreversible chains may not affected by this problem (see the first link below), but the problem is that you need to produce an irreversible chain with the right equilibrium distribution. 

Other thing you can try is using Probabilistic Cellular Automata (second link), which have an equilibrium distribution that is close to the Gibbs distribution asymptotically in the size of the state space . Since the latter is a class of chains with parallel update rule, the mixing time is sensibly smaller.

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You could look into nested sampling.

Instead of sampling from the posterior directly, it samples from a sequence of uniform distributions that contract around the peak. You can think of it as a cross between simulated annealing and MCMC.

Pros: 

• Excellent at exploring complicated and multi-modal posteriors
• Samples with very little a-priori information (no proposal matrices etc)
• Simultaneously computes traditionally extremely challenging integrals over the posterior, such as the evidence (normalisation constant) or relative entropy

Cons:

• No where near as established as traditional MCMC.
• If your posterior is very simple, it will likely never be faster than MCMC

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