From saddlepoint approximations you can get the approximation $P(X\ge x)\approx G_N(r_x)$ where $G_N$ is the upper tail probability of the standard normal distribution and $r_x^2=D_x=2\cdot[x\theta_x-m(\theta_x)]$ for $m(s)=\ln M(s)$ and $\theta_x$ the solution to $m'(\theta_x)=x$ (and the sign of $r_x$ determined by the $x$ being above or below the mean). The $D_x$ denotes the deviance, and the $r_x$ are (depending on context) sometimes referred to as deviance or likelihood residuals.
An alternative formulation of the same result for the case when $r_x\gg0$ (positive and not too close to zero) is $P(X\ge x) \approx \frac{1}{\sqrt{2\pi D_x}}e^{-D_x/2} =\frac{1}{\sqrt{2\pi}\cdot r_x}e^{m(\theta_x)-x\theta_x}. $
Note that this is an approximation, not a strict inequality as is Chebyshev's. The result can also be expected to behave better on continuous distributions than discrete, although it may work well on discrete distributions up to the neccessary errors due to the cumulative distribution having jumps which the approximation cannot capture.
In fact, the approximation produced by the reconstruction is log-concave (or close to being log-concave, not sure I remember correctly), so you can imagine that the actual distrubution is being approximated by a continuous log-concave distribution, and then the tail probability of this is estimated.
However, if the distribution is log-concave (continuous or discrete), I've seen this approximation work remarkably well, particularly at the extreme tails: not so quite so good very close to the centre.
Oh, a quick warning. My memory is a bit rusty, and I haven't had this morning's first cup of coffee yet, so I may have snuck in a typo here and there.
Here's one classic reference to this method: Barndorff-Nielsen, O., & Cox, D. (1979). Edgeworth and saddle-point approximations with statistical applications. Journal of the Royal Statistical Society. Series B (Methodological), 41(3), 279–312. http://www.jstor.org/stable/10.2307/2985061
There's also some (perhaps more readable) in this book if I remember correctly: McCullagh, Peter; Nelder, John (1989). Generalized Linear Models. Chapman & Hall/CRC.