Peter Svenonius (once again) asks the right question and Omer (once again) has interesting things to say about it (see here). Take a look. Here is my take on the issues. Please chime in with yours.
I agree with Omer that there may not be much of a consensus right now about how to deal with successive cyclicity in detail. However, so far as I can tell, there is general agreement that it has something to do with the PIC (this is the current analogue of the subacency principle, Bounding nodes and the domains they create).
As you know, there are two extant versions of the PIC, the more favored one being the one wherein a complement of a phase head is rendered inaccessible at the NEXT phase (usually when the next phase head is accessed). This comes close to coding the old idea of subjacent domain (as was early observed one has access to the domain one is in and the next one (i.e. no counting)). The strong version of a phase is less in favor, but it has some charms for it would force something like a "no edge skipping" requirement on the grammar (e.g. the old Rizzi idea that one could "skip" the most immediate Comp would be ruled out). There are purported arguments against this stronger version, but they never struck me as dispositive (and there are Legate style arguments against it). At any rate, there is consensus that phases should derive cyclicity.
How closely is this tied to features, uninterpretable or otherwise? Logically speaking, not that closely so far as I can tell. The issue of features is tied to whether movement is optional or obligatory. If Greed drives movement or uninterpretability then C features might be needed. Yes if Greed is strong and no if something like uninterpretability suffices to drive one to the phase edge as a last resort (Phase balance or Boskovic's take on the same idea). There is some evidence that intermediate Cs can have features, as we know. The generalization to all languages is a standard GG move. So, the idea is not empirically nuts. Of course, WHY this should be true is unclear in the absence of something like Greed, but then maybe this is an argument for a strong version of Greed.
This goes against the current fashion. It seems that nowadays movement is free again (free at last, free at last, thank the lord, free at last!). But then there is no requirement that there be intermediate features to drive movement, nor that there be uninterpretable features on WH to force it to move. The WH moves or it does not. If it does, then it must move to the intermediate C for PIC reasons. If it doesn't then no convergence. More specifically, what one needs are language specific requirements that force a given G to have a WH up top overtly in some languages (something like the old strong feature) or some kind of Rizzi Criterion that is fulfilled in G variable ways. This seems generally assumed in current technology, so no biggie here.
There is one last idea that has been tied to successive cyclicity: Chomsky's current idea about labels. Oddly, for Chomsky, the fact that there are languages where there appears to be agreement in non WH Cs with a moving WH is a big problem. Agreement should obviate further movement. Of course one can get fancier here with different features having different effects on labeling (and so movement), but this begins to hand code in the property we want explained (not a good thing to do).
That's the way things look from where I sit. So, there are several ways of getting edge to edge movement all involving the PIC in some fashion and thereby recoding the old subjacency criterion. I want to emphasize this: this is not a new explanation but a recoding of the old one (not that this is a bad thing).
Two last points: what is less clear to me is how this all hooks up with islands. Chomsky, it seems to me, is reluctant to take islands as G-real phenomena. He seems inclined to take the view he once criticized, viz: that islands are performance residues of complexity. I am skeptical myself, but it is a logical possibility. The Sprouse stuff has convinced me that it is likely false.
This leaves the question of how to code Islands in phases? That's easy (as anyone who has tried will attest). The problem is that the coding follows from nothing (why are D edges different from C edges? why weak PIC rather than stropping? Why transfer when next pause head chosen rather than next phase completed? Why C and D and v as phases? Why week vs strong phases?). In fact, the coding just recapitulates the machinery in classical Subjacency theory. Or, Minimalism has not given us any insight into the details of subjacency as of this date. So, islands stand as having no good deep explanation beyond the one that Chomsky already provided for Subjacency that I quoted in the body of the earlier post.
Second: we really would love to tie island effects with ECP effects as Barriers and Cinque-Rizzi tried to do. Why? Because the domains for bounding and ECP are so damn similar. It would be really odd (IMO too odd to be tolerable) were these driven by different mechanisms given that their domains are virtually identical. So, we need to find a way of finally addressing ECP questions within MP. In particular we need to find a way of unifying them in ways more conceptually acceptable than the Barriers/Lasnik-Saito theory did.
So island effects are currently no better understood within minimalist theory than they were within GB. The GB story can be smoothly translated into technologically acceptable minimalist terms, but doing so provides no insight. Moreover, some parts of the old theory, the ECP part dealing with adjuncts vs arguments and their differing locality conditions, really has not good minimalist counterparts (does anyone really thing gamma marking is part of FL/UG?). That's how I see things. You?