How to make smooth line movement?

Hello!

I’ve been searching it everywhere, but couldn’t find anything in the net. I want to make smooth line movement. I’ve tried to create motion, but line just moves, and I want it to move smoothly according to the path. What I want to achieve can be seen here:

http://www.noartistname.com/demo/

But that’s what I did so far - made movement out of several broken lines - how can I make more smooth movement?

Thanks,
Aleks.