Denote $\mu$ the mean ($\neq$ average), $m$ the median, $\sigma$ the standard deviation and $M$ the mode. Finally, let $X$ be the sample, a realization of a continuous unimodal distribution $F$ for which the first two moments exit.

It's well known that 

$$|\mu-m|\leq\sigma\quad (1)$$

This is a frequent textbook exercise:  

\begin{eqnarray}
|\mu-m| &=&    |E(X-m)|          \\
        &\leq& E|X-m|            \\
        &\leq& E|X-\mu|          \\
        &=&    E\sqrt{(X-\mu)^2} \\
        &\leq& \sqrt{E(X-\mu)^2} \\
        &=&    \sigma
\end{eqnarray}
The first equality derives from the definition of the mean, the third comes about because the median is the unique minimiser (among all $c$'s) of $E|X-c|$ and the fourth from Jensen's inequality (i.e. the definition of a convex function). 


Even though it is in general not true ([Abadir, 2005](http://www.jstor.org/stable/3533476)) that any unimodal distribution must satisfy either one of
 $$M\leq m\leq\mu\textit{ or }M\geq m\geq \mu\quad (2)$$ 
it can still be shown that the inequality 

$$|\mu-M|\leq3\sigma\quad (3)$$

holds for any unimodal distribution (regardless of skew). This is proven formally in [Johnson and  Rogers (1951)](http://www.jstor.org/stable/pdfplus/2236630.pdf?acceptTC=true&jpdConfirm=true) though the proof depends on many auxiliary lemma's that are hard to fit here. Go see the original paper.


At any rate, putting $(1)$ and $(3)$  together yields:

$$|M-\mu|\leq 3|\mu-m|$$ 

This could be extended to hold for the original claim (i.e. without the absolute values), provided that you restrict yourself to the class of distributions for 
which $(2)$ holds (among skewed distributions, the Beta, Log-normal and Gamma are three that satisfy $(2)$). See [Basu and DasGupta (1992)](http://www.stat.purdue.edu/research/technical_reports/pdfs/1992/tr92-40.pdf) for a more general characterization of skewed distributions that satisfy $(2)$.

 - [0]: The Moment Problem for Unimodal Distributions. 
N. L. Johnson and C. A. Rogers. The Annals of Mathematical Statistics, Vol. 22, No. 3 (Sep., 1951), pp. 433-439
 - [1]: The Mean-Median-Mode Inequality: Counterexamples
Karim M. Abadir
Econometric Theory, Vol. 21, No. 2 (Apr., 2005), pp. 477-482
 - [2]: Basu, S. and DasGupta, A. (1992). The Mean, Median and Mode of Unimodal distributions, A characterization. Technical report #92-40. Department of Statistics, Purdue University. September 1992.