I'm curious to find out where the viewpoint of higher categories may be useful so here is a somewhat vague question (which may or may not have a reasonable answer).
Given a triangulated category, one can consider the set of all possible t-structures on it. Simple examples where one can compute things by hand indicate that this is something complicated but not hopelessly so. See for example the paper http://arxiv.org/abs/0909.0552 by Jon Woolf which describes a three parametric family of t-structures on the constructible bounded derived category of $\mathbf{P}^1(\mathbf{C})$ stratified by a point and its complement. Some of these t-structures are more interesting than others and there is one that is the most interesting of them all since by taking the bounded derived category of its heart one gets back the triangulated category one started with. (For that t-structure the heart is the category of perverse sheaves on $\mathbf{P}^1(\mathbf{C})$.)
On the other hand, the set of t-structures on a triangulated category is interesting since there lurks somewhere the conjectural motivic t-structure whose existence implies Grothendieck's standard conjectures. See the recent paper http://arxiv.org/abs/1006.1116 by Beilinson.
On the triangulated categories page of the n-category lab website it says "Therefore, all the structure and properties of a triangulated category is best understood as a 1-categorical shadow of the corresponding properties of stable (infinity,1)-categories". See http://ncatlab.org/nlab/show/triangulated+category. Note that this is quite a strong statement, since it is referring to all, and not just some, properties and structure of a triangulated category.
So I'd like to ask: is there a higher categorical analog of a t-structure? More generally, how does the higher categorical viewpoint help one understand the set of all (or maybe all "nice" in an appropriate sense) t-structures on a given trangulated category, provided it is the homotopy category of a stable $(\infty,1)$ category?
upd: as Mike points out in the comments, the answer to the first question is yes and it is given by proposition 6.15 of Lurie's Stable Infinity Categories. The second, more "philosophical" question remains.
Best Answer
I wanted to contribute something because nobody's really explained why a t-structure on a stable $\infty$-category is the same thing as a t-structure on its homotopy category. It's not just because someone defined it as such in Definition 1.2.1.4 of Higher Algebra; it's because the most natural generalization is no generalization at all.
Roughly speaking, a t-structure on a triangulated category should have the following three properties:
h1. It determines a full subcategory $\mathcal D_{\geq 0}$ closed under the shift functor $\Sigma$.
h2. $hom(X,Y) = 0$ if $X \in \mathcal D_{\geq 0}$ and $Y \in \mathcal D_{\leq -1}$.
h3. Any object fits into an exact triangle, sandwiched between an object in $\mathcal D_{\geq 0}$ and $\mathcal D_{\leq -1}$.
If you think about the algebraic manipulations for which one utilizes t-structures, the natural generalization of a t-structure to a stable $\infty$-category ought to be as follows:
1$.$ It determines a full subcategory $\mathcal C_{\geq 0}$ closed under the shift functor $\Sigma$.
2$.$ $hom(X,Y)$ is contractible if $X \in \mathcal C_{\geq 0}$ and $Y \in \mathcal C_{\leq -1}$.
3$.$ Any object fits into an fiber sequence, sandwiched between an object in $\mathcal C_{\geq 0}$ and $\mathcal C_{\leq -1}$.
If $\mathcal C$ is stable, one defines a triangulated structure on its homotopy category $\mathcal D = ho \mathcal C$ by demanding that its exact triangles are precisely those arising from fiber sequences. Then we see immediately that (1)-(3) imply (h1)-(h3). That is, we have a map from "oo-categorical" t-structures on $\mathcal C$ to t-structures on $ho \mathcal C$.
What might be surprising is that the other direction holds, but it's actually an easy exercise to show that any collection of data satisfying (h1)-(h3) uniquely determines data satisfying the properties (1)-(3). In a nutshell: (h1) implies (1) obviously. (h1) and (h2) together imply (2) because you can write a fiber sequence for $X[i]$ and $X[i+1]$, then apply the hom functor to get a fiber sequence of mapping spaces. The long exact sequence of homotopy groups will prove that every homotopy group of hom(X,Y) vanishes. Finally, by definition of exact triangles for $ho\mathcal C$, (h3) also implies (3).
So we have an interesting phenomenon where the set of t-structures can't distinguish between two oo-categories with equivalent homotopy categories. This is one of the main reasons that Bridgeland stability conditions don't have an obvious generalization to see higher-homotopical structures beyond the homotopy category.