You are correct, you really can't dispose of that type of stuff with regular trash. A couple things I can think of
- Some communities will have a special day where you can bring odd things to dispose of: electronics, paint, chemicals, etc.
- Check to see if your community has a computer "recycling" company; we have one in town and you can just bring it to them and they'll take off your hands; they can break everything down and some parts can be recycled into new things
- I would say get rid of it via CraigsList, but if it's 5 years old, it may not be very desirable....but you never know! Monitors are not that expensive, plus the norm are the flat panels (assuming that yours is probably not flat panel!)
Good luck!