When will all eight planets in our solar system align? Ignoring expansion of the universe, entropy, decaying orbits, and interference from any bodies colliding with or otherwise interfering with their orbits, will the eight planets known planets in our solar system ever align?

What is the "period" of the planets; how often would they align perfectly? And based on their current positions, how far off into the future is their next theoretical alignment?
This is a low accuracy - yet simple - answer

It allows you to calculate only radial alignment configuration of the planets.

If you would like an approximation, let's say, you approximate the position of the planets as hands in a clock, you could work the math out by something like this.

Assume $\theta_i$ is the initial angle for planet $i$ at time $t_0$ - measured from an arbitrary but fixed position, and $l_i$ is the length of the year - in days - for planet $i$.

Then it resumes to solving this system of equations:

$$
x \equiv \theta_i \left( \ mod\ l_i\right)
$$

From here you would then simply apply the Chinese Remainder Theorem.

Finding the minimum x, will give you the angle that the planet that at $t_0$ had angle $\theta_i = 0$ would have travelled until an alignment configuration was reached. Asuming you choose Earth as the mentioned planet, then divide that angle by a complete revolution ($360^{o}$) and you will get the number of years for that configuration to be reached - from the $t_0$ configuration.

The different $\theta_i$ in degrees for all the planets on 01 Jan 2014 - you can use this as your $t_0$:

\begin{align}
Mercury &amp;\quad 285.55
\\Venus &amp;\quad 94.13
\\Earth &amp;\quad 100.46
\\Mars &amp;\quad 155.60
\\Jupiter &amp;\quad 104.92
\\Saturn &amp;\quad 226.71
\\Uranus &amp;\quad 11.93
\\Neptune &amp;\quad 334.90
\end{align}

Source

The different $l_i$ in days for all the planets:

\begin{align}
Mercury &amp;\quad 88
\\Venus &amp;\quad 224.7
\\Earth &amp;\quad 365.26
\\Mars &amp;\quad 687
\\Jupiter &amp;\quad 4332.6 
\\Saturn &amp;\quad 10759.2 
\\Uranus &amp;\quad 30685.4 
\\Neptune &amp;\quad 60189
\end{align}

Finally under an integer values aproximation and using this online solver for the system of equations the answer is $x = 4.0384877779832565 \times 10^{26}$ which divided by $360^{o}$ gives you roughly $$1.1218 \times 10^{24} \quad\text{years}$$

Edit 1

Just found this site you may like to play around with. It's an interactive flash application with the accurate position of the planets.

I also know that the all the information can be obtained from this NASA page and that is as accurate you can get, but it is just incomprehensible for me now. I will try to revise it later when I find time.

Also this book by Jean Meeus called Astronomical Algorithms covers all the fundamental euqations and formulas - it has nothing to do with programming algorithms though.

Edit 2

Seeing that you are a programmer, it could be worth for you checking out the NASA site I mentioned above, the data for all the planets can even be accessed via $\tt{telnet}$.
Or this Sourceforge site where they have implementations for many of the equations described in the book also mentioned above.