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 $2\pi i$.)