While this page describes sites where all cashier and account functions are anonymous, this is a different matter than actually playing anonymously. Still, if you are concerned about accessing games where no screennames are shown, then you do have options available.
At Nitrogbetting Poker, you can change your screenname whenever you want. Thus, you can enter in a new one before each session, and your opponents will have no way of knowing what your previous gameplay history was like. This is effectively a way to make your tables anonymous using your own initiative.
There are mainstream internet poker rooms that do not show users' screennames as a matter of course. Two of the most prominent are Ignition Poker and Bovada, which are part of the same online poker network. These two operators display next to each player just a seat number or, in the case of tournaments, a number indicating in what order that individual registered for the game. Bear in mind, though, that Ignition and Bovada are only anonymous on the gameplay end; you might still be required to prove your identity when cashing out or performing other account-related activities.