The speakers currently plugged into my computer are about to die, so I’d like to replace them. I’d like a decent set, but nothing too expensive–I’d balk at anything much over $100. Any recommendations?
I have a slightly older version of this Altec Lansing 3 piece speaker system:
and I have installed them for two or three customers. Everybody pretty much loves them.
Very good sound for a desktop set (although the subwoofer can be a little large for some desks). In my experience, very hard to beat for the price.
I bought some refurbished Klipsch Promedia 2.1 speakers and they look brand new. They have worked great and I could not believe the sound that comes out of 2 speakers and a sub woofer. It is also nice that they have inputs on the control speaker for headphones and audio in. You can plug in a portable CD player and it will play through the speakers without turning on the computer. They do not seem to have any at the moment but they do have the 2.1 GMX version. I have not heard them. Link
I’ve had the same set of Cambridge Soundworks for about 7 years now and they still sound pretty good. Unless the build quality has changed significantly over the last 3/4’s decade, I’d say that’s a pretty good recommendation. 
You CANNOT go wrong with the Klipsch 2.1 systems. Brand new they are about $150. But you get your moneys worth for that extra $50! I wouldn’t get anything less. I had the Altec Lansing 3 piece before I got my Klipsch setup. Altec Lansing, sorry, but never again!
There is no doubt the Klipsch system is better. I love Klipsch systems in general, and have used several of their systems as my primary sound system speakers for 30 years. Great stuff.
But the OP specifically requested under $100, and the Altecs are one of the best I’ve heard in that price range.
But if the extra $75 doesn’t kill you, absolutely, get the Klipsch!
http://www.epinions.com/cmhd-Speakers-All-Cambridge_SoundWorks_CSW_PCWorks
I’ve had this set for about 5 years, and have very pleased. It sounds better than my other stereo setup, both the speakers and sub are small, and both are very clear even at high volumes if you want to use it with your computer as a replacement of your home stereo.
Also, they were only 50 dollars when I got them, 5 years ago.
I should mention that on another tech message board a lot of the members there were praising the Logitech brand speakers. They are more affordable than Klipsch and have many different models that are in the price range mentioned in the OP. Here is a list from Newegg but Logitech are widely available at a lot of retailers.
I have the Altec Lansing 251 five-speaker plus base module system. I’m no purist audiophile, but i do like my music, and i’ve been very happy with the system. It cost me $99 about two years ago, and now i feel a bit depressed because it can be had at Newegg for $38.99 plus s&h.
Anyway, you can find some reviews of the system here, here, and here.
The reviews are all good, although it seems that true audiophiles might want a slightly more powerful, bass-heavy system. The cutomer reviews on newegg are mixed; some people are dissappointed, while others think the system is good value for money.
If this system is now selling for less than $40, you should be able to get a considerably better system for about $100.
I second this. It’s also easy to add speakers, mine has an extra pair of 5" mid-range.
I second the reccomendation for Logitech speakers - I have a set of z-640’s that I purchased for $50, and they sound wonderful.