> one can always define a t-structure on a stable model category as a t-structure on its homotopy category

This is a definition, but it's extremely unsatisfying and completely unenlightening.

Have a look at [this](http://arxiv.org/abs/1408.7003) paper, where my advisor and I prove that t-structures in a stable infinity-category correspond bijectively to suitable orthogonal factorization systems, called "normal torsion theories", i.e. to factorization systems  $({\cal E},{\cal M})$ where 

1. Both classes $\cal E,M$ satify the 2-out-of-3 property; this implies [CHK] that the category $0/{\cal E} = \{X\mid 0\to X \in\cal E\}$ is coreflective, and dually that ${\cal M}/0 = \{ Y \mid Y\to 0\in\cal M\}$ is reflective. These categories play the role of the aisle and coaisle of your t-structure.
2. The square
$$
\begin{array}{ccc}
\tau_{\ge 0}(X) &\to & X \\
\downarrow && \downarrow \\
0 &\to & \tau_{<0}(X)
\end{array}
$$
is a homotopy pullback and pushout.

More info also on the [nLab page](https://ncatlab.org/nlab/show/t-structure) and in subsequent [two](http://arxiv.org/abs/1501.04658) [papers](http://arxiv.org/abs/1507.03913).

> Why should this address your question? 

Well, we are able to characterize t-structures as a genuinely categorical gadget, living not on the level of homotopy categories, but in the "real" higher categorical world: since every model for stable categories has its own avatar of the semantics of factorization systems, you are able to speak about t-structures in every model:

* Quasicategories have quasicategorical FS, and hence "our" notion of normal torsion theory
* stable model categories, the setting you are interested in, have [homotopy factorization systems](https://www.math.rochester.edu/people/faculty/doug/otherpapers/Bousfield_Fact.pdf), and hence "homotopy normal torsion theories".
* DG-categories have [enriched factorization systems](https://ncatlab.org/nlab/show/enriched+factorization+system), and hence enriched normal torsion theories (I am not aware of anybody speaking of t-structures in enriched settings);

(I know, this is sketchy: it -especially the possibility to talk about these gadgets in stable derivators- is a work in progress with two colleagues)