When I say 'stopped working,' I mean that it wouldn't recognize a disc: I'd insert the disc, it would spin around a little, make some chugging noises, but then spit it back out. I think it starting doing this for CDs sooner than it did for DVDs (more powerful laser?), but I can't recall for certain. I don't do much with discs, and since it's a 4.5 year old laptop, I never got it serviced, which predictably landed me in a bind.
Anyway, that's when I stumbled across a solution:
You take a clean cloth (like a lens cloth for cleaning eyeglasses), drape it over something slim like a business card or a smooth plastic gift card, and plum the depths of your drive in an attempt to clean the lens. Supposedly it's just inside the left boundary, but I plunged my improvised cleaner in everywhere I could, as deep as I could get it. I didn't use a brutal amount of force, but I wasn't particularly gentle either. Result: now my drive works.
Fair warning: I'm sure it's possible to somehow do horrific damage to something in your drive. If you have the time, patience, and money, I encourage you to take your non-functioning drive to a professional. But if you like to try to fix expensive hardware yourself, then this hint is for you.
All the credit for this goes to Anar's post here.
[crarko adds: I suppose as a last resort, if a drive cleaning kit doesn't help, you could try this. Just be careful.]