Weight bearing exercises aside, doing plyometrics (jump training) is more beneficial without cushy sneakers as it actually reduces impact on the joints upon landing. Plyos are an important training method for athletes, as they increase explosiveness, but they're also beneficial to the average person because they dynamically improve coordination and balance.
Try this simple experiment: from a squat position, bend deep and then explode up as high as you can while wearing your sneakers, and pay attention how you land. Do you land with your legs mostly straight? Do you hear a thud? Now take off your shoes and socks and repeat the experiment, once again paying attention to how you land. Do you land with bent knees to reduce the impact and naturally bend back down into the squat position? Do you land on the balls of your feet instead of your heels? Do you hear nearly as much noise, or are you pretty much silent? Now think about which was the softer landing: with sneakers, or without?
The two biggest detriments to modern cross trainers, running shoes, etc. are the raised heel and compressible soles, which shift your center of gravity forward. This deactivates the posterior chain (the glutes and hamstrings) and places more impact on the knees and quads. This, in turn, alters the kinetic chain even further up the body by deactivating the anterior core (abs, obliques) and placing undue emphasis on the lower back. In sum, this all makes you weaker and more prone to injury. Similarly, a compressible sole acts as a shock absorber, thereby deactivating the calves - your body’s natural shock absorbers. Not only will sneakers increase the risk of injury over the long run, they will cause inefficient technique, ultimately causing you to generate less force. Doing box jumps barefoot or in minimalist footwear may feel strange at first, but will allow your body to more naturally execute both the jump and the landings without destroying your knees in the process. (Adapted from www.barefootconcepts.com.)
Read more advantages of going barefoot on the "Barefooting" page!