A few friends of mine recently purchased the Nike Free Bionic shoe for such workouts. They appeared to be a minimalist type of shoe and so at first I thought that the lack of support and cushioning would not work for my feet. Then I read up on them and figured, if I only wore them when I was doing strength training and TRX classes, they may be just the ticket! I went online and bought a very cute pair (they come in loads of fun colors)! I ordered them from http://www.footlocker.com and got them on sale for $79 + tax/shipping!
Black/Anthracite/Pink Foil | Width - B - Medium
Anyway, I tried these out at the gym the very next day for a strength class. They were perfect! Zero slippage when I was doing weighted reverse lunges. They are as light as air! I notice a huge difference when doing box jumps with comparison to when I wore my usual runners. These are very flexible shoes. As I've said before, they offer very little in the way of support so you couldn't/shouldn't wear them running (unless you are one of those people who can run in minimalist shoes without any injuries). I'm extremely happy with these shoes! At first when I put them on and walked around the house in them they felt really weird - they are nothing like my regular runners. As soon as I started working out at the gym, they no longer felt strange. I really had to pay no attention to my feet at all because for once they felt completely secure!
I highly recommend giving the Nike Free Bionic a try if you do a lot of strength training workouts! Again, I would use caution if you are looking for a multipurpose type of shoe (like a cross trainer) - not sure these will be what you are looking for or if they are right for your feet.
Cheers!
Jamie