FYI, I am using Tmux through the Mac OS X Terminal app.
I've used some of the above plus other sources to come up with:
bind k send-keys C-u \; send-keys C-k \; send-keys " clear && tmux clear-history" \; send-keys "Enter" \; run-shell "sleep .3s" \; send-keys "Up" \; send-keys C-u
The leading space in " clear && tmux clear-history" prevents the command from being written to the history file (providing you have your shell setup to treat leading spaces this way; google "hist ignore space" + the name of your shell for more info). I like to have this command not show up in my history since this is more inline with ctrl-k in the Terminal.
The first send-keys C-u and send-keys C-k will clear whatever is currently typed at the prompt to ensure that the " clear && tmux clear-history" is successful (e.g., if you've typed "ABCDEFG" at the prompt and have your cursor between the D and the E, this ensures that "ABCD clear && tmux clear-historyEFG" is not sent to the shell, which would fail).
The send-keys "Up" and last send-keys C-u clears out the last items from your shells internal history. Even with the trailing space mentioned above the internal history of the shell will include the " clear ..." line. Sending up and Ctrl-u gets rid of this.
Lastly, in iTerm I set ctrl-k to map to ctrl-a k (I have my tmux prefix set to ctrl-a), so I can type ctrl-k which is what my hands want to do from so many years of doing so. I do this, by going to iTerm > Preferences > Profiles > Keys and adding a shortcut to send the hex code "0x01 0x6B". There's a great article here which gives more info on using hex codes with tmux and iTerm: http://tangledhelix.com/blog/2012/04/28/iterm2-keymaps-for-tmux/
That pretty much gives me ctrl-k with tmux. The only thing that still kinda nags at me is that the real ctrl-k without tmux doesn't have problems if you currently have something typed at the prompt and will preserve what you've typed while clearing the screen. As mentioned, this approach needs to clear what's typed so the " clear ..." command doesn't fail. But it's pretty damn close!