Whenever that happens, you'll need a video converter, and Handbrake is probably the best choice due to two main reasons: It is very powerful and it is totally free.
When you run Handbrake, you see a very intuitive interface where you'll choose the input file - Video_TS folder - and the destination folder. Once you have selected them, you only have to choose the device where the file is going to be played from the list you see on the right: Apple TV, iPod, PSP, PS3, iPhone, iPod Touch, XBOX 360,... and Handbrake will do the rest.
If you are a novice you'll find it easy, ad if you are an advanced user don't hesitate and modify some settings to make the output video suit your needs. From DVD to iPod or PSP in a pair of clicks, just amazing.