Out of all the literature and history I have read (which I will admit is fairly limited), I feel as though I have never read anything about ancient Romans or members of the General Assembly "going for a run" or "lifting weights." I know that there are probably many logical reasons, such as their lifestyles were much different, i.e. they didn't sit in a cube all day and eat junk food. But i'm curious at which point in history we decided that our daily routines didn't provide enough physical activity, so we needed to simulate it ourselves.
It depends on whether you would classify it as 'working out' or 'training' (or if you believe there is a clear distinction between the two) but Roman troops conducted regular marches with full kit, learned to swim, practiced with weighted weapons and equipment (they supposedly weighed twice as much as regular weapons and shields) and were selected based on physical characteristics including "strength" and specifically "his chest should be broad, his shoulders muscular and brawny...his arms strong." You can find details in the De Re Militari by Vegetius, available here: http://www.digitalattic.org/home/war/vegetius/index.php#b103
The public baths were also a place for exercise, with a courtyard for various exercises and of course a pool for swimming. There are also many tales of legendary feats of strength and endurance in the ancient world and certainly athletics were widespread, with training involving some exercises that fundamentally are similar to training today, albeit less sophisticated. A good example that is very well known is the legend of Milo of Croton carrying a bull on his shoulders every day until it (and his shoulders) were fully developed.