Using padic analysis, Dwork was the first to prove the rationality of the zeta function of a variety over a finite field. From what I have seen, in algebraic geometry, this method is not used much and Grothendieck's methods are used instead. Is this because it is felt that Dwork's method is not general or powerful enough; for example, Deligne proved the Riemann Hypothesis for these zeta functions with Grothendieck's methods, is it felt that Dwork's method can't do this?

The premise of the question is not correct. Dwork's methods (and modern descendants of them) are a major part of modern arithmetic geometry over $p$adic fields, and of $p$adic analysis. You could look at the two volume collection Geometric aspects of Dwork theory to get some sense of these developments. Just to say a little bit here: Dwork's approach led to MonskyWashnitzer cohomology, which in turn was combined with the theory of crystalline cohomology by Bertheolt to develop the modern theory of rigid cohomology. The $p$adic analysis of Frobenius actions is also at the heart of the $p$adic theory of modular and automorphic forms, and of much of the machinery underlying $p$adic Hodge theory. The theory of $F$isocrystals (the $p$adic analogue of variations of Hodge structure) also grew (at least in part) out of Dworks ideas. To get a sense for some of Dwork's techniques, you can look at the Bourbaki report Travaux de Dwork, by Nick Katz, and also at Dwork's difficult masterpiece $p$adic cycles, which has been a source of insipiration for many arithmetic geometers. In some sense the $p$adic theory is more difficult than the $\ell$adic theory, which is why it took longer to develop. (It is like Hodge theory as compared to singular cohomology. The latter is already a magnificent theory, but the former is more difficult in the sense that it has more elaborate foundations and a more elaborate formalism, and (related to this) has ties to analysis (elliptic operators, differential equations) that are not present in the same way in the latter.) [For the experts: I am alluding to $p$adic Hodge theory, syntomic cohomology, $p$adic regulators, SerreTate theory, and the like.] 


I agree with Emerton's answer (and had similar thoughts, but since he is a leading expert in this field, it's better if the answer comes from him). I would say that if anything the opposite is true: nowadays people (especially those of an arithmetic bent) are more interested in $p$adic cohomology than $\ell$adic cohomology and the former is viewed as richer and more difficult than the latter. Thus the importance of Dwork's work could scarcely be higher. Flipping to the other side of the Weil conjectures, I think it is also not quite fair to say that Deligne proved the Riemann hypothesis "with Grothendieck's methods". I know you mean that he used Grothendieck's methods ($\ell$adic cohomology) rather than Dwork's methods ($p$adic analysis) but it doesn't do justice to the range of new ideas that Deligne brought to the table (as well as the ideas that were left on Grothendieck's table, to Grothendieck's eternal consternation). Flipping back again, note that the Weil conjectures (and even parts of Deligne's significant generalization "Weil II") have since been proven completely by $p$adic methods, c.f.:
This is another sign that the "Dwork school" is going strong in contemporary research. 


If you want to actually compute zeta functions of varieties over finite fields, then the padic methods are often more efficient. See e.g. the work of Kedlaya. 

