I’ve noticed when Googling “Christians and tattoos” that cogent arguments for the acceptability of Christians tattooing themselves under biblical mandate seem to be sparse. When recently challenged about my own tattoos by a self-recognized “Messianic Jew” (one who believes in Christ but attempts to retain the Jewish traditions associated with early Christianity) I answered with the following:

Note: I had initially misquoted the scripture as saying “do not cut your skin as the pagans do”. 

“26 Ye shall not eat with the blood; neither shall ye practise divination nor soothsaying.

27 Ye shall not round the corners of your heads, neither shalt thou mar the corners of thy beard.

28 Ye shall not make any cuttings in your flesh for the dead, nor imprint any marks upon you: I am HaShem.

29 Profane not thy daughter, to make her a harlot, lest the land fall into harlotry, and the land become full of lewdness.

30 Ye shall keep My sabbaths, and reverence My sanctuary: I am HaShem.

31 Turn ye not unto the ghosts, nor unto familiar spirits; seek them not out, to be defiled by them: I am HaShem your G-d.”

-Lev. 19:26-31 (Tanakh; Jewish Virtual Library)

You are correct, I fully misquoted the verse in confusion with something that my professor, Rabbi Spiro, commented on the passage. He subscribed to the school of thought that the above passage is, in context, about the Hebrews requirement to stray away from pagan practices. Upon a second and third reading of this conservative, Jewish translation of the text I think, though my quoting of the scripture was off-base, that my original argument stands. 
Consider three things:
First, the scripture is grouped into pairs of instructions that the redactors deemed similar to one another. In this, we see the cutting of the flesh and adorning of “tattoos” as being in the same group. It could be argued that the practices were extensions of each other as the other groupings (namely verse 27) seem to be either repeated for specificity or simply redundant. If they are grouped together, then you could more specifically argue that this calls against the tattooing of ones body for the mourning of the dead (a pagan practice and one that is often practiced today through “memorial tattoos) and as none of my tattoos are in remembrance of anyone, then I stand vindicated.

Secondly, the framing of the single verse (28) by other pagan practices (specifically of mourning but also of temple harlotry) sets up the context of these verses as supporting the idea that the intention was to separate the Hebrew people from their pagan neighbors. To quote Pastor Chuck Gerwig, “The “tattoo” marks described in Leviticus 19:28 were clearly related to false religious practices. The word translated tattoo in our English Bibles is the Hebrew word “qa aqa”, this word appears only one time in the Bible, here in this passage Leviticus. The word “qa aqa” means literally “to cut” but taken with the surrounding words indicates a cutting that left a mark imprinted in the skin. This could have been a form of branding, scarring, cutting or a process where ink was inlaid into the skin; there is not enough data to fully define exactly what this word meant. However we translate the word “qa aqa” though, in this passage, it is certainly used in the context of cultic religious worship. The prohibition against “qa aqa”, (translated tattoo) was to keep the Israelites from being involved or affiliated with cultic worship practices.” In this, my original argument stands.

Lastly, if we accept that this is a cultic practice, and that it’s prohibition by G-d is meant to ensure that we as believers use our bodies as an outward sign of devotion to Him, then the argument that both circumcised and uncircumcised people can be saved comes into the equation. We are no longer under the Old Testament Law as Christians, circumcision is no longer required. This is brought out in a number of New Testament passages, among which are the following: Acts 15; Galatians 2:1-3; 5:1-11; 6:11-16; 1 Corinthians 7:17-20; Colossians 2:8-12; Philippians 3:1-3. As these passages proclaim, being saved from our sins is received through faith in Christ to save us from our sins, and it is this act of turning from our sin and self-righteousness and turning instead to reliance upon Christ’s finished work on the cross that makes us “circumcised of heart” and that the works of the flesh accomplish nothing. Thus, if someone who has tattoos already accepts Jesus and professes him or herself a Christian, then it is no longer that only Pagans have tattoos and the need for separation ceases to exist.

In closing I would say this, the strongest argument given in the context of this passage is to mark yourself in memorial may be prohibited, but the marking of ones skin simply for decoration (not unlike make-up or the styling or coloring of one’s hair) just does not seem to hold argument for those of us who find uncircumcised freedom in Christ.