> I'm curious if this is common with laptops vs. desktops. I keep wanting to
> get a laptop for convenience, but this doesn't sound very convenient.
From my experience, overheating is definitely a common problem with laptops. Any computer can overheat, which is why all are built with fans. However, with laptops, because they are so small, there are only a limited number of places the fans can be placed. Most have fans on the bottom and side, which means that your comp might want to suck in cool air from underneath it -- causing problems if a hot lap is underneath it! A coolmat device like one poster bought would fix the problem. Another way to enable cooling is to always use a laptop on a hard surface like a desk (failing that, you can try putting a big book or something on your lap -- not perfect, but OK).
In my experience, some laptop brands/models are just badly made and will always have overheating problems no matter what. My gf (who I think bought a "lemon" b/c her laptop is always in the shop) has a laptop that runs its fans almost constantly (and very loudly!). This is not a good thing b/c sooner or later the overworked fans will probably give out and fry her computer. If you are thinking of buying a computer, I would discourage purchasing the lower end models from places like dell, toshiba, etc. They have crappy cheap parts in them and will tend to have a lot of problems -- more problems than a desktop computer, usually. It's always also good to buy the best warantee you can for a laptop, since they are so problem-prone.
All that being said, laptops are totally convenient and lots of fun! There are tons of good reasons to buy one -- just be careful!