Phemie is correct, that "as I do" is grammatically correct. However, "as me" is used colloquially by most native English speakers far more often than "as I do." So while it is technically incorrect to use "as me," it is more natural in speech. I still suggest that you use "as I do" in proper writing, however.
Personally, I wish that native English speakers did use correct grammar in speech. But the truth is that most do not. So to speak like a native, you sometimes have to speak incorrectly.