I was also running into the same issue (DNN 7.0.3), but the solution mentioned earlier by Daniel Comp did the trick for me:

<span data-bind="text: ProfileProperties().Telephone"> 

Don't forget to copy this into the template and alternating template. Also insert a value in your userprofile. The only struggle is to get the label as prefix in my template though. Only the value of the specific label is visible now.

'TelephoneText' as mentioned earlier by Cathal Connolly doesn't work for me.

Anyone got a solution for this?

I'm afraid, Cathal, that your comment does not produce accessible profile properties that are accessible.

"visite the profile page and click on my profile, and then do a view source and search for "ProfileViewModel" - you will be able to see the knockout model and read the variable names and values, hopefully that will help identify if you are using the wrong field names"

While I find the right field names, I am still not able to access them. They simply show up as blank, even though the knockout JS has access to their values, it doesn't spit them out in the output code on the page.

I tried to access Biography and the browser says that it doesn't exist as an object.

There is a ProfileProperties observeable in the member directory code, returning an array of profile properties. You should be fine with using something like this:

data-bind="text: ProfileProperties()['Company']"