The way academic research is communicated is obviously not ideal in lots of way. You have lots of publicly-funded research that's not available to the general public (and often involves waiting for an inter-library loan even if you work for an academic institution). The turnaround between papers being submitted and being published is often incredibly slow - I've seen plenty of cases of it being several years. Research usually has to be straitjacketed into a particular length and format. And the amount libraries pay for journals is quite scary when you consider that nobody pays the authors or referees.
It's also quite easy for cliques to emerge where you've got a small group of people all refereeing eachother's papers and it's not always obvious who are the best two or three people to referee a particular paper. Sometimes the people who could make the most insightful comments aren't the people who've already worked on that exact niche, especially in more interdisciplinary research.
Journals however do have three uses that I can see.
First, they perform some sort of filtering - you don't have to wade through the absolute rubbish which is often not that quickly detectable especially in more technical fields, but at the same time they do give a bright unknown which an amazing result a chance to get noticed. They smooth the A-list effect to a certain extent.
Secondly, the role they perform editting is actually more valuable than I think lots of people appreciate. From my limited experience refereeing mathematics papers I know that good mathematicians aren't necessarily fluent in English and I'm very grateful for the service that editors perform. It would be a shame if important research got lost because a researcher couldn't afford an expensive technical translator or editor.
Thirdly, journal publication is used as a means of assessing researchers. If you can't understand their research totally, or don't have time to, pretty much the only benchmark you have is whether they have published in good journals or not. Even if you don't like this, then any solution has to assume that there is a demand here. If researchers get jobs by publishing in journals, that is very understandably what they will do.
I must admit that unlike newspapers and many magazines, I don't really see a great need for printed versions of journals. It can occasionally be nice to sit and browse through a whole issue of a journal, but on the whole it's specific papers that you're after. If you get rid of printed versions, you can also get rid of the idea of issues which frees you up much more. Open Access is also obviously good though you still have to deal with the issue of who should pay for what.
It is with refereeing where I would personally like to see more experimentation. I don't see why you couldn't say publish all submitted papers somewhere and allow anyone who wishes to comment as well as having designated referees. Then after a fixed period of time decide whether to 'publish' an article based in an intelligent manner on the comments received, with maybe some opportunity for authors to make changes based on the comments. It might take some tweaking to get this to work, to make sure that it didn't become a populariy contest and that only comments that provided sounds arguments for or against publication were considered. You'd probably also have to allow at the option of a degree of anonymity on both sides, to reduce problems of bias and people being loathe to criticise something written by someone who might have an influence on their career. But I think it'd be interesting to see other models of refereeing tried out which might reduce turn-around time, and improve the quality of papers generally.