There are many differences between the atheist civil rights struggle and other civil rights struggles, notably those of racial minorities, women and homosexuals.
The discrimination and oppression faced by atheists, while unacceptable, does not rise to the severity of that suffered by others.
Atheism is also in no way an ineluctable characteristic. To the extent that "choice" has any meaning at all, one chooses atheism (or metaphysical presuppositions that entail atheism) just as one chooses her religion. That some characteristic is ineluctable is itself a prima facie case for equal civil rights for those who have that characteristic. Obviously, atheists cannot avail ourselves of this argument.
Most importantly, though, atheists believe—and claim rational justification—that atheism is not just equal to theistic religion, but better: Atheism is held to be true and theism untruthful: false or bullshit. It's bad enough to be discriminated against; it's even worse to be discriminated against because we're actually right. One civil right that atheists demand is the right to actively criticize religion as untruthful.
This unique component of the struggle for atheist civil rights deserves to be explicit. Because we are advocating such a right, it's critically important that we make a pre-concession of strong secularism to religious believers:
I personally hold, and I exhort every atheist—presently and in perpetuity—to hold that legal and social coercion should never be employed to suppress mere falsity.
Normally, I'm not in favor of pre-concessions: One should—in principle—come to the bargaining table asking for everything one wants, and make concessions as a part of the negotiating process. However there are circumstances where pre-concessions are warranted.
First, there is the historical reality of the First Amendment, at least in the United States, which guarantees the freedom to exercise one's religion. This principle has worked reasonably well for a couple of centuries, and preserving it intact seems greatly preferable to opening a can of worms by attempting to change it.
Second, there's the present reality that most people are in fact believers. Negotiations cannot proceed if one party is demanding not only equality, but the reversal of oppression. This principle is important even when a neutral analysis might hold that a reversal of oppression was justified, such as in the Chilean and South African reconciliation movements.
Third, there's an important matter of underlying principle: The truth is the truth precisely because one need rely only on rational argumentation to establish the truth; it is not necessary to rely on coercion. While it's possible to force people to believe things, true or false, it's very difficult to do so, requires the imposition of enormous suffering, and the benefits are trivial and transitory. Even more importantly, it's physically impossible to force someone to think rationally, and it's logically impossible to force someone to think freely.
It is more important to support freedom of thought than it is to enforce the truth.
For this reason, the atheist struggle for civil rights must explicitly have two components: Secularism, the legal right to hold and express our beliefs from an a priori position of equality, and the rational case that atheism is actually true. We demand the legal right to present our case, for which we will absolutely and irrevocably concede that, even if we are successful in proving and persuading the majority of people to rationally accept atheism, we will never in turn coerce or oppress those who for any reason merely dissent.
 Laws against libel and slander target not only the falsity of statements, but also their malice and proximate cause of immediate harm.