It's been years since I lived in Seattle, but Belletown is a good area to look in for this kind of place. Also, under the viaduct near Pioneer Square is the OK Hotel. Good music, plenty divey. I don't really remember how cheap it was, but if I was there it must be pretty cheap! :^)
I'm a big fan of the Canterbury, on 15th ave in Capitol Hill. Vaguely medieval theme, cheap, strong drinks, pool tables, and greasy food. Basically everything you can ask for in a dive bar, unless you're looking for live music.
Ah, the Canterbury! When I lived in Seattle, a group of us would get together there on Sunday evenings for "church" (beer plus [mostly] political and philosophical banter). Church moved to the Elysian (on 23rd?) after some management shake up (this was almost 10 years ago). I'd forgotten about it, but I second bob's vote for the Canterbury!