There are only 4 ways to loose oil.
1. Leak to the outside - shows up somewhere outside the engine, on the undercarriage, on the ground, etc. I had this problem. My rear main crank seal was leaking and it showed up on the oil pan bolt heads. You do have to crawl under it and check for yourself, or get it up on a rack. You may need to do some cleaning and then recheck it to be sure where it's coming from.
2. Leak inside to the water jacket - shows up in the radiator water, usually a head gasket issue
3. Burn it due to bad rings - shows up on accelleration away from a stop.
4. Burn it due to bad valve guides/seals - shows up after coasting down a long hill with the overdrive locked out (engine braking) and when you hit the gas at the bottom of the hill you get smoke
You should be able to detirmine the problem based on these tests. How much it costs depends on which one it turns out to be. It shouldn't be that hard to find, 1qt per 1k miles is a lot. If you're burning it the tailpipe will have a dark oily residue instead of being dry.