To my knowledge the pumping lemma is *by far* the simplest and most-used technique. If you find [it][1] hard, try the [regular version][2] first, it's not that bad. There are some other means for languages that are far from context free. For example undecidable languages are trivially not context free.

That said, I am also interested in other techniques than the pumping lemma if there are any.

EDIT: Here is an example for the pumping lemma: suppose the language $L=\{ a^k \mid k ∈ P\}$ is context free ($P$ is the set of prime numbers). The pumping lemma has a lot of $∃/∀$ quantifiers, so I will make this a bit like a game:

1. The pumping lemma gives you a $p$
2. You give a word $s$ of the language of length at least $p$
3. The pumping lemma rewrites it like this: $s=uvxyz$ with some conditions ($|vxy|≤p$ and $|vy|≥1$)
4. You give an integer $n≥0$
5. If $uv^nxy^nz$ is not in $L$, you win, $L$ is not context free.

For this particular language for $s$ any $a^k$ (with $k≥p$ and $k$ is
a prime number) will do the trick. Then the pumping lemma gives you
$uvxyz$ with $|vy|≥1$. Do disprove the context-freeness, you need to
find $n$ such that $|uv^nxy^nz|$ is not a prime number.

$$|uv^nxy^nz|=|s|+(n-1)|vy|=k+(n-1)|vy|$$

And then $n=k+1$ will do: $k+k|vy|=k(1+|vy|)$ is not prime so $uv^nxy^nz\not\in L$. The pumping lemma can't be applied so $L$ is not context free.

[1]: http://en.wikipedia.org/wiki/Pumping_lemma_for_context-free_languages
[2]: http://en.wikipedia.org/wiki/Pumping_lemma_for_regular_languages