IIRC, there is a canon prohibiting self-mutilation. My apologies for not being able to reference it off the top of my head (contrary to popular opinion, most monks don't sleep with copies of The Rudder under their pillows). The trick is: figuring out what exactly that is. Is a tattoo mutilation, or enhancement? That's the entire crux. If a tattoo is mutilation, then it's out. If it's not, then it's not prohibited.
Considering how many matushka's have pierced ears, and I wouldn't dare accuse them of mutilating their bodies (especially if you've ever had to deal with a fiesty matushka!), there seems to be some room for interpretation on what is mutilation and what isn't. And it is a common, historical and (IMO) venerable practice amongst the Coptics and Ethiopians.
That being said, amongst Russians there seems to be an assumption that a tattoo marks one as a member of organized crime (the vor v zakone), or at least seriously anti-social like some sort of skin-head gang, so they tend to be less sympathetic towards it.
There was also some scant evidence that the early Celtic Christians continued the practice of tattooing themselves, just with Christian symbols rather than pagan ones, IIRC. But, I don't think the scholarship is entirely conclusive about that one. But that's why they were called "Picts"; because they were the "pictured" men, or men with "pictures" on them.
I'm pleasantly surprised that no one has brought up the Old Testament prohibitions against tattoos. I always cringe when people do that, to be frank. It betrays a serious lack of understanding of how the Old Testament laws are fulfilled in Christ and frankly don't apply to Christians. You can't just "cherry pick" OT laws that you like (usually for other people to follow!). You either follow all of the Law (and be Jewish BTW), or none of it. If you want to prohibit tattoos based on Leviticus 19:28, then you also have to have fringes on your clothing, a parapet around your roof, keep kosher, keep the Saturday Sabbath and can't shave the sides of your head, amongst other things. The Law is an all or nothing kind of thing. It's either all 613 laws, or none.
Now, I personally think one should probably get the blessing of one's spiritual father before getting a tattoo. If one already has them, well what's done is done, and unless one is rather wealthy, can't be removed. But, I wouldn't get anymore (I have two) without my spiritual father's blessing, and they would also have to be deeply symbolic and meaningful, rather than just a "pretty picture".
Just my thoughts though. Your mileage may vary.