You can:
https://www.bitwage.coI have been an advocate of a Bitcoin Credit Union because I thought that this was the only way to get paid in Bitcoin by giving your company a direct deposit number. But it looks like this company has figured out how to do it by going through payroll services (I think).
I signed up but don't have my phone for authentication but I will definitely try it out and let you know how it goes. I would love to get paid in Bitcoin. With a Xapo card and localbitcoins for cash I can drop my banks very soon.
Update:
I was able to get my account set up and log in (I had some complications because I live in Germany working for a US company). Their customer service is great, my problem was sorted out quickly and easily, if you have problems click the little chat window on the bottom right of the screen.
Once I got my profile set up it asks for my company name, company website, my company's payroll company and the payroll company's website. Most of us do not know these things offhand and will need to work with your HR. I will be working on this to see how easy it is from there and let you know how it goes.
Update 2:
I e-mailed my HR and asked what payroll service they use because I needed it for bitwage.co so I can get paid in bitcoins. They responded after what appeared to be a long e-mail chain bouncing around all over the payroll department asking if anyone can help out, finally responding that I need to speak with my specific HR representative. Fortunately, in their lengthy e-mail chain they are talking about the Payroll company they use. So I submitted that to bitwage.co and I am currently in the verification process.
A bit of advice when asking your HR department about your payroll company: don't tell them it's for bitcoin or bitwage. Just ask for the name of the payroll company. You can use google to find the website from the name. You can tell them why after they give you the name but mentioning bitcoin is like going to the train station and saying you're going on vacation to Hawaii and what is the best route to get there.
Update 3:
My Verification level 1 was completed. I was able to do my Verification level 2 by submitting a photo of my ID and proof of address. Once that was verified I was able to get to the part where I enter my Bitcoin Address. With my Bitcoin address entered they provide me with a bank account and routing number to give to my payroll for direct deposit. My direct deposit website at my company allows me to create a fixed dollar amount account and a net account for anything over that amount. I stuck a test amount of $500 in my fixed dollar amount to go to the Bitcoin direct deposit. Now I will wait to see if it goes through.
Update 4:
Just got an e-mail saying my payment went through, $500 at $317/BTC -> 1.56992012 sent to my Bitcoin address.
Next, I will convert all of my paycheck to bitcoins. I can covert BTC->USD as needed.