Are there any good tricks to construct a heart of a t-structure? (I'm thinking on the derived category of coherent sheaves of some variety)

I'll start with the only one I know. If $(T,F)$ is a torsion pair on an abelian category $A$ then you can form the *tilt* inside $D(A)$.

By definition, the tilt is complexes in $D(A)$ such that the minus one cohomology lies in $T$ and the zeroth cohomology lies in $F$ and all other cohomology vanishes. Unfortunately this method is only good for hearts concentrated in two degrees.

An example of such a torsion pair is if you take $Ab$ the category of abelian groups, $T$ torsion groups and $F$ free groups. In any abelian group you can find the largest torsion subgroup and the quotient will be torsion free. This can be generalised to arbitrary integral domains and from that to arbitrary integral schemes, although the geometric meaning of the tilt (if there is one) escapes me.

Another example is Bridgeland's category of perverse coherent sheaves, which can be defined as a tilt. I would be very interested in generalisations of Bridgeland's category. There is also a notion of perverse coherent sheaf by Bezrukavnikov, but from what I understand it is only useful with Artin stacks, as the jump in the dimension of the coarse space given by stabiliser groups is what allows interesting perversities. Comments on these latter perverse sheaves would also be very much appreciated, as I don't really understand the construction.