Graphical models relate to HMMs as a graph relates to a path. This means that there is an explosion of possibilities, which is both a good thing and a bad thing. The good news is that you can build rich models for generating sequences of random observations that have desirable properties; the bad news is that the parameters of your model are no longer so easily interpreted. In fact, the identifiablity problems that you experience with HMMs are greatly multiplied.
Still, this is a subject which is here to stay, and if you are interested in HMMs, you will want to keep a close eye on the developments in this rapidly growing subarea of statistics.
One good place to start is with Kevin Murphy's 1998 piece: "A Brief Introduction to Graphical Models and Bayesian Networks."
I will post other resources as I sort them out. Feel free to tell me about resources (print or hypertext) that I should link to here.
CMU Graphical Models Reading Group's Selected Resources. This is a shopping cart of resources on graphical models. It is almost a whole course, so some poking around is need if you are just getting started. There are some broken links, but you can probably just google your way to the originals.
Bernt Schiele's ETH course on Graphical Models and Bayesian Nets.
There is a cacophony of software for Graphical Models. This is organized (somewhat telegraphically) by Kevin Murphy's Table of Features of Graphical Model Software.