You've already got a lot of advice, but I would like to suggest an alternative strategy for minimizing the potential for bias without worrying employers that you are trying to hide something. Rather than scrubbing your name and references to your gender from every part of your resume, try to ensure that anyone reading your application has to learn a little bit about you before learning your gender. For instance, if you apply by emailing your resume with the email body as the cover letter, send the email from firstname.lastname@example.org and don't mention your name at the beginning of the email, but put your name in your resume and consider signing the email with your full name.
I assume here that most people who might be hiring you are implicitly but not explicitly biased. In many cases, implicit bias happens instantaneously*. So, if a hiring manager sees an email from Emily Smith, they may be predisposed to read your cover letter in a negative light. On the other hand, if they don't learn your name and gender until they get to the bottom of the letter or open your resume, they will have the chance to consider the case you make for yourself before being prejudiced. By providing your real name at the end of the letter and/or on your resume, you may be able to avoid the possible suspicion of an unnamed candidate that others have mentioned.
I don't know whether or not this strategy will work for you, but I can provide some anecdotal evidence from the academic world, where a similar strategy is quite common. Many women (and a few men) will write papers and abstracts with just their first initials, but include more details about who they are through profiles and affiliations. The idea is to mitigate immediate bias from reviewers and readers, but allow interested parties to figure out the author's identity easily. Colleagues of mine who do this believe it has helped them, though it's obviously impossible to tell.
*This assertion is based on my understanding of the Harvard Project Implicit and related research using the same methods.