Hopefully others on the forum jump in on this as well, but I have to say I think it's really unfortunate that this PT gave you this advice and continued the spreading of what I like to call the myth of 'fat burning zone'.
There are aspects of what this PT told you that are true. Namely that at a lower heartrate a greater PERCENTAGE of the calories being burned will come from fat.
But there is information missing in the statement, that leads to the myth that you'll burn more fat at a lower heart rate. Namely, at a lower heart rate much fewer calories are being burned than at a higher heart rate. So at a low heart rate less total fat calories and less total calories from blood sugars are being burned.
Walking is a great activity... and it certainly is much better than no activity. But there is no question, minute for minute, mile for mile, running will burn more total calories and more total calories from fat than walking will.
There is an article in our library section that discusses heart rate zones. And although it wasn't written specifically to discuss weight loss, it does speak to this very common question. You may want to check that article out.
In your question you ask about 3 hour of walking a week vs. 1 hour of running a week. Now of course, this is a more complicated question because you've changed the amount of time you're committing to walking vs. running.
So, let's look at one technique to estimate the differences in calories burned by these two activities.
The typically accepted value of METs for "Walking for exercise, 3.5mph, brisk pace" is 3.8 METs. Running at a 10min/mile is listed as 10.0 METs. (For more information on METs and how they related to calories, check out the article Burned by Calories Burned in our library.)
If you accept these MET values, and using the formula cal=MET*Hours*Kg then walking 3.5mph for 3hours would burn slightly more calories (11.4 calories per kg) than running 6mph for 1hour (10 calories per kg). But if you ran for 30 minutes each day, you'd easily burn more calories at (15 calories per kg).