Do I have to run a disk integrity check after a kernel panic? Relatively new to Macs, I'm used to Windows wanting to verify HD data integrity after an unexpected shutdown. I know they use different file systems, but do Macs need to do anything like this?
Mac OS X's default file system is HFS+ Journaled. This means that before files are written, the data is first written to a special log called the journal. In the event of application or system crashes, when the machine re-starts it will re-run all the remaining entries in the journal to bring the file system back into pristine order. In theory anyway.

That said, it is still probably a good idea to use one of the included utilities to verify your filesystem.

You can run fsck in the Terminal or use Disk Utility to check your hard drive if you want. After a Kernel Panic or power outage, this is probably a good idea.

Disk Utility is the simpler option, just log in as normal and go to /Application/Utilities/Disk Utility and select your boot drive and choose "Verify". You can also run Disk Utility from your Mac OS X DVD that came with your computer (you will need to do this if you want to repair, it can't repair the boot volume if you are running the computer on it).

If you want to run fsck, you will have to do it from the startup terminal.  I'm a little fuzzy on the details, since it has been years since I last used it, but on startup hold down ⌘S during startup to boot in single user mode. Then run fsck to run the diagnostic. You should also look at man fsck to look at the manual.

TL;DR
No, you don't generally need to. But it's not a bad idea to run a utility.