This issue is largely settled today--- you cannot go to another disconnected universe, but you are either trapped in the black hole, or reemerge in this universe. The reason is the no-information loss property. If material could go between disconnected universes, information about the state of this material would be permanently lost to the other universe. Hawking in fact has recently admitted that this idea was the main motivation for his initial proposal that Hawking radiation is perfectly thermal, and does not encode information about infalling stuff. This would be a definitive sign that the black hole is linking to a disconnected universe
The reason people expected to be able to travel to another universe is because a generic rotating black hole has three regions. The easiest way to see the structure is to consider a charged black hole, which is qualitatively the same, but spherically symmetric. The metric has f(r) in the dt part, and 1/f(r) in the dr part, with $f(r) = 1 - a/r + b/r^2$, where b is the square of the electric charge, and a is half the mass.
The zeros of the time component of the metric tell you the position in r of the horizon. There is an outer horizon and an inner horizon (the third zero is at negative values of r, and is unphysical). The outer horizon is an event horizon--- once you cross, you are forced inward. The inner horizon is a Cauchy horizon, once you cross this horizon, you see the singularity at r=0.
Gravity becomes repelling in r inside the inner horizon. Then the particle falling in is shoved outward, to cross the Cauchy horizon going the other way, and then it crosses the event horizon going the other way, but considering the event horizon as a white hole horizon now. Classically, each of the regions is disconnected, there is no identification between them. So that you go from our universe to the interior, then to the inner part, then to the interior of the white hole, then out into another disconnected universe.
In the 1970s, the white hole and black hole horizons were considered distinct, so that a black hole would be one thing, and a white hole another. But Hawking himself noted that thermal equilibrium states are invariant under time reversal, so that by time reversing a black hole in equilibrium with radiation, you find that a black hole and a white hole are actually the same, once quantum mechanics, but in different perspective.
The full infalling process is not understood, because no-one has traced infalling matter going through a black hole in AdS/CFT (although it is just out of reach--- people are working on it). The result (I believe) is that the infalling matter does the classical circuit, and comes out of the event horizon at a time that is 1/hbar, so that the transit time diverges classically. But this is not supported by any detailed calculation, just by vague futzing around in AdS/CFT models. The only calculations in the literature make the claim (equally unsupported) that the black hole absorbs the stuff that falls in irreversibly, without spitting it out. I am pretty sure this is not true, it spits it out in this universe, if so, sometimes CPT reflected, so it comes out anti-matter.
I speculated that this could be the reason for the anomalous anti-electron signal observed recently by satellites near the galactic center. A rotating astrophysical black hole could be a copious antimatter source.