The most well-understood examples are the ones you mention: Axiom A diffeomorphisms and Markov maps of the interval, since these can be modeled by SFTs.  Note that "Bernoulli" refers to a particular choice of invariant measure; the SRB measure for an Axiom A attractor (or the ACIP for a Markov interval map) is Bernoulli, but other invariant measures, such as periodic orbit measures, need not be.  More generally, given a system modeled by an SFT, equilibrium states for Hölder continuous potentials are always Bernoulli.  (Recall that these are invariant measures $\mu$ that maximize the quantity $h(\mu) + \int\phi\,d\mu$, where $h(\mu)$ is Kolmogorov-Sinai entropy and $\phi\colon X\to \mathbb{R}$ is the potential function.)

Beyond this, I know of basically two classes of examples that are known to be Bernoulli; both are "non-uniformly hyperbolic" in some sense.  The first class contains systems that can be modeled by a Young tower or a countable-state Markov shift, where the measure is an equilibrium state for some sufficiently regular potential function; this includes the case where the measure is SRB, in particular when the measure is smooth.  For example, this includes any positive entropy equilibrium state for a Hölder continuous potential on a surface diffeomorphism, thanks to recent work of Omri Sarig ([JAMS 2013][1], [JMD 2011][2]).

In fact, if the measure is smooth and has non-zero Lyapunov exponents, then you don't need to build a Young tower or a countable-state Markov shift; for measures like this ("hyperbolic" measures), Bernoullicity was proved by [Yakov Pesin in 1977][3].  I think this extends also to SRB measures (possibly due to Ledrappier?), but I don't have the reference off the top of my head, I'll check and add it here if I can remember it.  In particular, this includes Liouville measure for geodesic flow of a compact surface of genus at least two without focal points (of course this is continuous-time rather than discrete-time as you asked for).


  [1]: http://www.ams.org/mathscinet-getitem?mr=3011417
  [2]: http://www.ams.org/mathscinet-getitem?mr=2854097
  [3]: http://www.ams.org/mathscinet-getitem?mr=466791