The problem is that just because you're using more fat as fuel doesn't mean you're actually burning more fat off your body. Burning fat is more about overall calorie expenditure, not just about the type of energy your body is using for your workout. Another problem is that you may not be able to workout as long or as hard if you're hungry, which means you may end up burning fewer calories than if you'd eaten something and worked harder.
There are other benefits to eating before your workout:
It can boost recovery and strength gains
It can help you sustain longer, more intense workouts
It can help you avoid low blood sugar, which can make you feel dizzy or nauseous
It can make your workouts more enjoyable (since you're not thinking about eating the whole time)
The bottom line is, we each have to find a system that works for us. You may be fine doing cardio without a meal in the morning, but strength training may require more fuel to really challenge your muscles. The best answer to this is to do what works for you. Don't go hungry just because you think you're burning more fat...after all, if you cut it short or lower the intensity because of low energy, how much fat are you burning anyway?
If you do eat before a workout, make sure you give your body time to digest. The larger the meal, the more time you'll need. But, if you choose a light snack (100-200 calories) and stick with higher carb fare, you can probably exercise after about 30-60 minutes. Pre-workout snack ideas:
Banana (or other type of fruit)
Energy bar or gel