Shifter moving

I have an 02 Tacoma Double cab w/automatic. My shifter moves back and forth while driving, especially on acceleration. Is this normal or is it a bracket broken or something. Any one else have this problem?

Yeah I think it's normal. I have a 98 std cab w/ automatic and both my shifter for my transmission and my t-case shifter move around on acceleration and also going over large bumps in the road. It looks like it's either going to pop into neutral or downshift into 2nd. It's really weird and has been doing it ever since I got the truck. But I have 120,000 miles on my car and no problems whatsoever. I love my truck and I have all the confidence in the world with it.

Same thing here. Have a 98 Extnded Cab 4X4 V6 Auto, and it's done that about a dozen times, and I've got about 130,000 miles on mine. Seems to happen when I hit a large bump and accelerate at the same time, it's almost as if the wheels were off the ground, grabbed hold, and pulled the gear shift into neutral, it's never went the other way. I've noticed it after I see the RPM's start to jump, and don't feel the forward motion anymore. I took it to the dealer back in 98 when I got the truck, and they couldn't get it to repeat the action, so they said it wasn't anything to worry about. Hasn't affected the truck in any way. Good Luck, and I hope you enjoy your truck as much as I've enjoyed mine.

My DC is brand new, it does it too. all Toyotas do to some degree. It is simply torque induced or can be caused to a lesser degree by suspension movement.
If the V8 motor in the new 4Runner had a stick mounted directly to the trans/trans-x then you would see some SERIOUS movement.