This is an important issue that I don't feel is addressed strongly enough by atheists. Many atheists think it's not a big deal because you never have to recognize God and can choose a secular affirmation.
Not good enough. The mere allowance of religious inclusion in affirmation is problematic. It can create an inappropriate bias among the jury.
Imagine your average American jury listening to testimony from several people who all swear in the name of God and then hearing testimony from someone who takes a secular affirmation that doesn't include God. Do you really think that none of the jurors will weigh this testimony differently?
Giving a secular affirmation will be likely to hurt the credibility of testimony heard by religious people. Beyond that, there's the additional problem of them having to request a separate affirmation. To many mind-sets this will paint the person as whiny and un-American. "What, the traditional oath isn't good enough for you?"
Allowing any mention of gods puts unnecessary personal information before the jurors that is likely to taint their perception of the testimony. It aides religious biases and bigotry and offers no benefit. Allowing for secular affirmations does nothing to alleviate this and, when used, may well make the problem worse.