First, let's understand that running is a form of conditioning, which can be done either aerobically or anaerobically. There are several ways to condition, and some are compatible with weightlifting and some do work against it. The first point I want to make is that you do have a mistaken supposition: that running will burn muscle. The two articles referenced in IronStrong's Conditioning forum will help shed some important light on the subject.
Metabolic Conditioning Energy Sources:
- Muscles pull from Adenosine Triphosphate (ATP) for all their energy needs
- When demand is low (aerobic), energy is sustained by oxidative pathways such as burning fat
- When demand is high (anaerobic), energy comes from glycolysis such as burning sugars. This happens without oxygen.
- The aerobic pathways are still engaged when the glycolitic pathways are engaged (i.e. you are still getting aerobic work)
- Longer period aerobic work (jogging) processes the lactate buildup of the anaerobic pathways through the aerobic work.
In short your choices are burning sugar and fat. It's not an either/or proposition, when you are doing anaerobic work you are also burning fat just as you would with your aerobic work. The differences then are in the adaptations on where your muscles look for its main source of energy. Purely aerobic work will cause your muscles to adapt towards being more efficient at being aerobic--which is the opposite of what you need for weightlifting. Purely anaerobic work still incorporates the necessary amount of aerobic activity to support the higher demands.
In reality, when it comes to conditioning you want to condition for the types of stress you would normally encounter, or at least would need to be prepared for. A 40 yard sprinter would need a different conditioning plan than a relay racer. A martial artist would have different needs than a football lineman. In short the conditioning should support the work you need.
The "Death by Prowler" article by Matt Reynolds is very informative, and provides a few tips and approaches to customize your conditioning program for your ultimate goals. While he focuses on the prowler as the tool of choice, there is no reason it won't work for sprints, sledgehammer conditioning, etc.