Just to be a little more specific, you do not need to keep LR3 installed on your computer, but the terms of your license with Adobe require you to not transfer LR3 to another person if you have upgraded to LR5; if you think of it that way, you should "keep" LR3 in that sense. The license terms allow you to have Lightroom (any version numbers) installed on any two devices (which could mean a Mac computer and a Windows PC, or a laptop or desktop, or any combination along those lines). If someone else were to attempt to register the same LR3 key you're supposed to still have to run your LR5 upgrade, either you or the other person could get blocked in that attempt.
Before upgrading, it's always a good idea to download and try out LR5 to make sure it runs well on the computer you want to use it on--some devices that can run LR3 well can't run a later version well, and if that applies to your hardware, it's best to figure it out before you've paid for LR5.