Handwriting absolutely should be continued to be taught in schools. The rise in the use of computers in the classroom is not a good thing for the educational process. Computers can be used for good things such as power point presentations and other visual aids. However, they should not be used for taking notes. The level of retention is not nearly as high when typing when compared to handwriting notes and the temptation of twitter, facebook and the rest of the internet is likely too tempting for most students. If you put them in front of computers full-time, you're just asking them to be distracted.
Some firsthand experience: As an undergrad, I never ever used a laptop in class (in fact, most professors expressly prohibited them from being used). I took all of my notes by hand. In law school, I experimented with using my laptop and ultimately ended up using a laptop to take notes in the vast majority of my classes. (That, however, is because law school sucked and law professors don't really teach you anything about the material so much as they teach you that you better do your work before class so you don't look like an idiot if they call on you.)
I can say that when I wrote my notes by hand, I retained way more information than I ever did just typing notes on my laptop. Part of that may be because I can type way more words per minute than I can write (and I'd say that that is the case for most people) and therefore I just copied down most of what the professor said. When I wrote my notes by hand, I had to be more selective so I parsed out what mattered and what didn't which made me much more engaged in a lecture or a discussion.
Working as an attorney, I started taking notes by hand but then tried computerizing the process. I made a form that I could easily fill in for each of my cases. While I was able to input the data more quickly, I found that I was able to recall a great deal more about the cases on which I took notes by hand than about those about which I took notes on my computer. As a result, I now went back to taking notes exclusively by hand.
As for the cursive thing, I use cursive but I think that it's a matter of personal preference. I just find it much more fluid which makes it easier to take notes. They should still teach it in school so then children have the option of using it but they can also abandon it after the 3rd grade like the majority of the population chooses to do.