You could use a third-party privacy service. There are some that are not associated with a registrar.
You could have an attorney act as agent.
Or, you could simply get a P.O. Box, a voice-mail phone number, and use a company name rather than your personal name. (Of course, the company name would be traceable.)
The latter is what I've elected to do with all of my domains. It's cheaper than paying for privacy service on a number of domains. If you can get one (they are in short supply at many post offices) a U.S. Post Office Box (as opposed to a private one) is a great bargain. And voice-mail numbers are practically free.
I don't have any need to hide my ownership - just don't want nutcases showing up on my doorstep. Especially with the "about us" wiki scraper sites that are popping-up like mad that also scrape your WHOIS info, and plot it on a convenient Google map!
I recently moved some domains from one registrar to another, where they had been using a privacy feature. I just changed the info to the P.O. box and voicemail before transfering. I didn't have the 60-day problem, as the losing registrar didn't have that restriction.
Of course, there is no such thing as totally private anyway. You are required to give your registrar accurate information, and they are required to release it under legal discovery or subpeona.
Although I have used a privacy service in the past, I now think that looks scammy, and I think it would be completely fair for search engines to penalize sites for using them. (Particularly, say, for e-commerce sites.) Not saying that they do - but that I would think it legit if they did, and think that perhaps they should.
It seems you do not need to put a personal name as registrant or in any of the contact roles - look at the WHOIS info for major sites. You will see "Web Master", "Domain Administrator", etc.