Are there "motivic" proofs of Weil conjectures in special cases? This is a question meant as a first step to get into reading more on Weil conjectures and standard conjectures. It is known that the standard conjectures on vanishing of cycles would imply the Weil conjectures. So, are there proofs of Weil conjectures in special cases using partial results on the standard conjectures? If so, which cases, and what are the references?
Background: Borcherds mentions here that Manin proved a few special cases in higher dimensions using motives.
 A: Weil's proofs for curves and abelian varieties essentially use special cases of the standard conjectures, and the framework of the standard conjectures (like many other conjectures of a motivic nature) is suggested by trying to generalize the abelian variety case (or if you like,
the case of H^1) to general varieties (or if you like, to cohomology in higher degrees).
Deligne's proof for K3 surfaces uses a motivic relation between K3s and abelian varieties (which is most easily seen on the level of Hodge structures) to import the result for abelian varieties into the context of K3 surfaces.  This is not so different in spirit to Manin's proof for unirational 3-folds, except that the relationship between the K3 and associated
abelian variety (the so-called Kuga-Satake variety) is not quite as transparent.
[Added, in light of the comments by Donu Arapura and Tony Scholl below:] In the K3 example, it would be better to write "a conjectural motivic relation ... (which can be observed   rigorously on the level of Hodge structures) ...". 
A: Of course, there's Serre's Analogues kählériens de certaines conjectures de Weil, Annals 1960, where he deduces an analogue of the Weil-Riemann hypothesis  over $\mathbb{C}$ using standard facts from Hodge theory.
This is technically not an answer at all, but I thought I'd mention it since I had the
(perhaps mistaken) impression that this was partly the inspiration for the standard conjectures.
The other more relevant comment is that one can give an elementary proof of the Weil conjecture for any smooth variety whose Grothendieck motive lies in the tensor
category generated by curves. I should explain,
especially in light of Minhyong's comments, that this could be understood as shorthand
for saying the variety can built up from curves by taking products, taking images, blow ups along centres which  of the same type, and so on. Actually, for such varieties, the
Frobenius can be seen to act semisimply. I think this open in general. So perhaps there's
some value in this.
A: The paper by Manin is MR0258836 
Manin, Ju. I.

Correspondences, motifs and monoidal transformations. 
Mat. Sb. (N.S.) 77 (119) 1968 475--507 where he uses motives to prove the Weil conjectures for unirational projective 3-folds. 
