Run one day, lift for upper body the next day, HIIT the third day, abs and light cardio the fourth day, lower body lift the fifth, rest day and go for a bike ride on Sunday.
Sites like fitnessblender.com can help you get there. (I've been mentioning them in these threads because I love them, the changes in my body and most importantly, in what can I do with my body have been nothing short of amazing in a relatively short amount of time)
I occasionally run/cycle/hike/shovel snow/etc. as well, but working for the three weight lifting workouts per week. I'm not faithfully "doing the program", but incorporating elements that work for me into my life.
I agree rest days should be rest days. I've paid the price of not resting a couple of times too many already.
As a tall guy, it feels much more natural and 'safe' for me. I've also switched to doing more warmup sets for my deadlift with increasing weight ranges. If any of those lighter weights feel 'off' that day, I abort the heavy sets. I also avoid testing my 1RM frequently...I'm content to add 5 pounds every time I can get through 3x5 at my heavy sets weight. I'm in my mid 30s, so safety is the name of the game for me.
Did some resistance training a few years ago - felt good, lost fat while doing it.
Shifted to cardio for a few years with an employee sponsored cardio-centric wellness program. Experienced a gradual fat gain and knee and back pain, although I was running 20-30 miles per week, using far too much time per week.
Shifting back to weights the last few months. Not being very rigorous, but doing the exercises in "Starting Strength". Back pain* and knee pain now very infrequent. Strength is increased. Some muscle definition improvement (though lots of excess fat still).
* Lower back pain actually made me very hesitant to do deadlifts. I started with the bar and the very lightest weights and gradually progressed, being VERY cautious about strain on the lower back. I'm only dead lifting ~200 lbs at this point, but my back is much happier.