Here is a general fact (due to a bunch of people in the late 60s, early 70s):
A connected space is (weakly equivalent to) an infinite loop space if
and only if it admits an action of an $E_\infty$-operad.
Let me try to unpack this statement. An $E_\infty$-operad $\mathcal{O}$, first of all, is an operad all of whose spaces $\mathcal{O}(n)$ are contractible.
(Sometimes also want to require that the action of the symmetric group on $\mathcal{O}(n)$ be free; this is important if you want the theory of $E_\infty$-algebras to be homotopy invariant. For instance, if you take your $E_\infty$-operad to consist of a point in each dimension, then sure, an algebra over that operad -- that is, a topological abelian group -- is an infinite loop space -- but those algebras tend not to be interesting, and certainly don't model all infinite loop spaces. Note that topological abelian groups are always weakly equivalent to infinite products of Eilenberg-MacLane spaces.)
Now being an algebra over an $E_\infty$-operad is a way of saying that your space is as close to a commutative monoid as possible. That is, there's a multiplication law (pick any point in $\mathcal{O}(2)$ to get a map $m: X^2 \to X$), it's homotopy associative (that's because the two $m(m(\cdot, \cdot), \cdot), m(\cdot, m(\cdot, \cdot))$ are both 3-ary operations coming from $\mathcal{O}(3)$, which is contractible). Moreover, there are higher coherence homotopies (infinitely many) which are conveniently packaged in the operad: that's one of the things operads do efficiently!
One example of a coherence condition is the following. So we know that $m(x, m(y, (m(z, w)))$ and $m(m(m(x, y), z), w)$ are both canonically homotopic (as maps from $X^4 \to X$). But there are two different ways we could make the homotopy go. We want a coherence homotopy between those two homotopies. This is the analog of the MacLane coherence axioms on a monoidal category; you want the various iterated identifications one can make between iterated multiplication laws to be all homotopic.
I'll also mention a weaker (and easier) result:
A connected space is (weakly equivalent to) a loop space if and only if it admits an action of an $A_\infty$-operad.
An algebra over $A_\infty$-operad is something which is supposed to be as close to associative as possible. The standard example is the little intervals operad. If you take $\mathcal{O}(n)$ to consist of the space of imbeddings of $n$ intervals in the interval, then that acts on $\Omega X$ for any $X$. (How? Use these embeddings to compose a bunch of loops.) The point is that being an $A_\infty$-space, rather than simply a homotopy associative H space, is the data you need to construct a classifying space. (The whole story essentially begins with the theory of classifying spaces; it's what shows you that any topological group $G$ is the loop space on $BG$.) You can do this either by first strictifying your $A_\infty$-space into an actual topological monoid (yes, you can do this, essentially because the associatve operad is still "reasonable" insofar as it is acted on freely by the symmetric group; Berger and Moerdijk's paper on homotopy theory for operads), and then take the usual classifying space. Or you can do it directly, e.g. as Segal does it in the paper I mention below.
The result for $E_\infty$-operads and infinite loop spaces is supposed to be more or less the following: take iterated classifying spaces. I don't think that's how May does it (though I don't understand May's construction that well at the moment), but apparently Boardman and Vogt prove it that way.
A very intuitive and fun paper on this sort of thing is Segal's "Categories and cohomology theories." Segal introduces his own form of delooping machinery, which in fact implies the result about $A_\infty$-spaces that I described above.
Finally, let me say something about exactly how
$E_\infty$ spaces are like abelian groups. There is a general theory of algebras and commutative algebras in a monoidal $\infty$-category, due to Lurie (developed in DAG II and III, now in "Higher Algebra"). The definition works out so that "commutative algebra" really means "homotopy commutative algebra up to infinitely many higher homotopies" (as it always does in $\infty$-land). So an associative algebra object in spaces is an $A_\infty$-space, and a commutative algebra object is an $E_\infty$-space. I think one of the motivations of higher category theory is to say efficiently what "up to coherent homotopy" means. For instance, you can think of an $\infty$-category as a "topological category where multiplication is associative up to coherent homotopy."
Best Answer
As Qiaochu points out, I'm simply wildly wrong in believing that a bounded $t$-structure is determined by its heart -- although if you fix an ambient triangulated category $\mathcal T$ it is true that the bounded part of any $t$-structure on $\mathcal T$ is determined by its heart.
As alluded to in the comments, there is a recognition principle for determining when a $t$-structure is the derived category of its heart, given in Lurie's Higher Algebra, Prop 1.3.3.7 (used to spectacular effect by Gheorghe, Wang, and Xu). There is always a functor from the derived category of the heart to the original triangulated category. Lurie gives a recognition principle for when this functor is fully faithful and identifies its essential image. In the case of spectra, it boils down to asking whether $Ext_{Spectra}^\ast(\mathbb Z, \mathbb Z)$ vanishes for $\ast > 0$. Which it doesn't, because there exist nontrivial stable integral cohomology operations. Stable integral cohomology operations are not that familiar, so I'm a bit more comfortable $p$-completing and observing that the forgetful functor from $H\mathbb F_p$-modules to $p$-complete spectra is not fully faithful because the Steenrod algebra at $p$ is nontrivial.