"Should I register multiple extensions of my domain name?"
The answer to this question depends entirely on how you intend to use your name.
If you are planning to have a commercial site, an on-line store or service, it is probably a good idea to protect your name. There is nothing so frustrating as working hard and putting out a great deal of expense to develop and market a fantastically successful .com web site, only to find that someone has registered your name as a .net or .org, hoping to reap some of the benefits of your success. Especially when you consider the minimal costs of acquiring the other versions of your name when compared with the total cost of getting that killer web site up and going.
You may also want to consider that when you do establish your successful web site, your competitor may try to benefit from your accomplishment by registering a name very similar to yours with a slight variation to take advantage of common typos and anticipated misspellings. And don't forget a plural form if you are selling a certain type of product (e.g. computer.com and computers.com)
Larger firms have taken to registering derogatory versions of their names as well (example: companystinks.com) to prevent disgruntled customers or former employees from putting up injurious sites. This can get into quite a list of names, though.
If you are an internet hobbyist and are just planning to put up an informational site (and we truly salute you for it!!) or a site designed to network with people who share your interests, you may not need or even want to register multiple extensions. You may rather want to give someone else a chance to put up a good site with a good name.
It's all up to you.