What is the origin of this Western movie theme?
Earliest reference I can find :
“Bob Hope in “Road to Utopia,” trying to look tough as he enters a saloon and orders a lemonade, then, realizing what he’s done, growls at the bartender: “In a dirty glass!””
There’s a funny variation on this joke in Rustlers’ Rhapsody (a funny movie overall) when the hero orders “a tall glass of warm gin with a human hair in it”.
“I’m sorry, we don’t serve drinks in dirty glasses.”
“Well, you did the last time I was in here!”
You know, this just sounds like a Bob Hope joke. 
I think I heard it in a Woody Woodpecker cartoon once, uttered by arch-nemisis Buzz Buzzard.
I saw Cary Granite do it on The Flintstones, once. His drink was rocks on the rocks.
Tough customer, that Granite.