What would be the best way to say "not mine" or "this isn't mine" in Japanese?
I sometimes go to cafes and use the internet for long periods of time to get work done online. Often the Japanese staff will walk around and pick up trash around the cafe and sometimes there is trash on the floor or on a table near me. I want to be able to easily explain in a short sentence that the trash doesn't belong to me. Would I just say something like: "俺のじゃない"? That is what I usually say, but maybe there is another way because the woman just kept saying "はい"(yes), so maybe she had no idea what I was saying.

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I would probably say "それは私のではありません" (Sore wa watashi node wa arimasen), meaning "it isn't mine." Or "それは私のではない" (Sore wa watashi node wanai). 
Although you could use ore/boku.

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I agree with Orsinia-san, in the cafeteria or something, the stuff asks you "is it yours?"(in polite Japanese). If it isn't yours, you can say いいえ or ちがいます. (Of course, these two together is also correct. )

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Thanks everyone.

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I think she understood your saying correctly and her "はい" means "I see." or "OK, no problem." in this situation.
So it seems you do not have to change your way of saying.