MacBook :: CD / DVD Drive Won't Read Discs

First, my MacBook is a little more than a year old and is no longer covered under warranty, so I would like to try any possible fixes (like cleaning the drive) before having to buy a new drive. Here's my problem: when I put a CD or DVD into the drive, it spins around and clicks a lot but never seems to really catch, and after about a minute or so of this, it ejects itself. It never shows up on my desktop, so the computer is unable to read anything from the CD or DVD. I know the problem is not the CD or DVD, because I've tried the same discs on other machines and they work fine. This problem came up on me gradually. It started happening a few months ago, and maybe only 1 out of 10 or 20 discs would it do this. Then it became a third of the time, then half of the time, and now it's about 99% of the time I try to insert a disc. I was able to get one disc to work a few days ago, but when I ejected it and reinserted it, it didn't work the second time.