SoM - great thoughts there. Funny you should mention the Saz because I have one of those. Lark in the Morning (in California) has a website with lots of exotic World instruments. The one I got is six stringed, three double courses, with a banjo style body and long, thin neck with nylon fishline tied on for frets (apparently in the old Turkish tradition for this instrument).
Like a lot of exotic instruments, this is more tourist quality. I got a book on Saz playing from LITM also, and it was of course in Turkish. I had someone semi-translate for me. But the biggest issue is the frets are not lined up the same way as the book. Luckily I have an Intellitouch clip-on tuner - I plan to move and remove some frets to make it match the book.
ANYWAY!!!
The point is, often these instruments are completely alien but as you say, can inspire you to play your normal instrument in ways you hadn't considered.
The baglama Saz and the Oud are made for the middle eastern microtones. Basically, from what I have heard, between every sharp/flat and the adjacent note, there is a quarter-tone.
Trip Wamsley's IT'S BETTER THIS WAY has a great microtonal song, I think it's track 2, and it may in fact be Michael Manring doing the playing, I am not sure. But if you have a fretless (Alembic) bass it would be an inspiration.
The oud is the ancestor of the lute and guitar and most other Western fretted instruments.
Good luck with it!
EffClef