For some strange reason, my DVD/CD burner suddenly stopped burning or even recognizing CD-R's. It has no problem burning DVD's or CD-RW's - but it does not want to recognize CD-R's anymore. Where should I start in diagnosing this glitch? Has anyone ever heard of this happening before?

Take off the cover to see if the belt is off on the bottom.
However, this is usually an equipment failure which cannot be repaired. CD drives have three or four componets -- CD-R, CD-RW, DVD, and DVD-RW. Any one of the four can fail independently.
Years ago, you could get four or five years out of a CD player. Now you are lucky to get much more than a year. Some are now only warranteed for 90 days, and none have a warranty longer than a year.
Luckily, they are easy to replace. You can get a combo CD-R, CD-RW, And DVD drive at Wal-Mart, Best Buy, Frys, Circuit City and others for $39 to $49... and they will last about a year.
Avoid Sony, Pioneer, HP, Lacie, and HCT, but the LG and Samsung are good. The Plextor and Yamaha at much higher prices are superior products.