The ingredient in the Beilinson and Bloch--Kato conjectures is a motive (over ${\mathbb Q}$, say). If we take the integral cohomology of this motive (mod torsion, say) we get an integral lattice. If we take some kind of Neron model, and take the algebraic de Rham cohomology of this, we get a second integral lattice. Now computing the determinant of the pairing of one of these on the other, we get a transcendental number, well defined up to a unit in ${\mathbb Z}^{\times}$, i.e. a sign.
This should give you an idea of how one can attach a canonical period to a motive, and is the basic idea underlying the construction of periods for motives. (Since one doesn't have Neron models in general, this idea is just heuristic as it stands, but I think it gives the right idea. If you apply it to ${\mathbb G}_m$, you should recover the period $2\pi i$.)
EDIT: I should point out that the above really is just a heuristic, explaining how there are two ways of getting integral structures in cohomology: in singular cohomology, one just takes integral cycles (i.e. "true" cycles on the motive, with no funny coefficients), and in de Rham cohomology, one takes algebraic differential forms that are defined over the integers, like the Neron differential $dx/2y$ on an elliptic curve with minimal Weierstrass equation $y^2 = f(x)$.
To actually get the correct periods for a given $L$-function, one has to do a little more manipulation than I indicated; e.g. for an elliptic curve over ${\mathbb Q}$, one will integrate the Neron differential over the a basis for the real integral cycles (i.e. the cycles that are fixed by the action of complex conjugation on $E({\mathbb C})$; these are rank one subgroup of the cohomoloy of $E({\mathbb C})$). But hopefully what I wrote above gives some intuition for what is going on.