I'll flesh out the consequences of Gerald's comment in a (CW-ed) answer.  Lindenstrauss, Olsen, and Sternfeld showed [in 1978][1] that if $S_1$ and $S_2$ are compact metrisable simplices such that the extremal points of $S_i$ are dense in $S_i$ for $i=1,2$, then there is an affine homeomorphism from $S_1$ to $S_2$; the unique (up to affine homeomorphism) compact metrisable simplex with the property that its extremal points are dense is called the *Poulsen simplex*.

In that same paper, it was shown that the Poulsen simplex has the property that its set of extremal points is arc-connected.  Since $\mathcal{M}$ is a compact metrisable simplex whenever $X$ is a compact metric space and $f\colon X\to X$ is continuous, and the extremal points of $\mathcal{M}$ are precisely the ergodic measures $\mathcal{M}^e$, it follows that $\mathcal{M}^e$ is arc-connected whenever it is dense in $\mathcal{M}^e$.  In particular, the strong specification property introduced by Bowen implies that periodic orbit measures are dense in $\mathcal{M}^e$ ([Sigmund 1974][2]), and since such measures are ergodic, this implies that $\mathcal{M}$ is the Poulsen simplex, and hence $\mathcal{M}^e$ is arc-connected, whenever $(X,f)$ has strong specification.

So that's not quite as constructive a proof as the approach following [(Sigmund 1977)][3] as suggested in Andrey's answer and the comment following, but it's certainly simpler to write down based on existing results.


  [1]: http://www.ams.org/mathscinet-getitem?mr=500918
  [2]: http://www.ams.org/mathscinet-getitem?mr=352411
  [3]: http://www.ams.org/mathscinet-getitem?mr=447528