Designing a streambed for roof runoff

I bought a copper rain chain to replace one of our downspouts, and hung it on a corner where the ground slopes down through a narrow (12') area between our garage and fence. Now I'd like to put in a water feature, but not your traditional pond. I envision a runoff channel that would be dry most of the time, but it would be like a mini-stream when rainwater is pouring off the roof. I want to direct the water down toward our neighbors' enormous spruce, about 25' away.

Can anyone give me any design ideas or practical tips on how to do this? Do I simply dig a ditch and throw in some pretty rocks? Or does it need a liner (old tires or something)? How would I make a little cascade? Keep in mind that the only time it rains very much is May-June. We get 11-12" of precip a year, which makes this a mountain desert.