Another recommendation for Sublime Text 2 (its in beta, but already more stable for me than Textmate ever was to have switched over to full-time).
It has a similar auto-complete behaviour to Textmate that you mentioned you needed, triggered by the tab key by default. However, it doesn't cycle through the different matches with this key - it actually just selects the best match. As pointed out by koiyu above, if you hit ctrl-space it will open the auto-complete drop-down with all the matches that you can choose from.
More info on its completion behaviour can be found here
Other notable features if you are interested:
Sublime Text is highly configurable. Everything from the default key-mappings mentioned for the auto-complete behaviour above, down to the colour of the tabs in the interface can be easily changed by modifying the JSON-format configuration files. A lot of things don't have a pretty GUI at the moment but it was easy enough for me to get in there and tweak everything I could possibly want -- ST is a tweaker's dream.
Textmate-compatibility. As an ex-Textmate user, I love that ST is compatible with Textmate snippets, language files and colour schemes. I couldn't find a CMake bundle in Sublime Text by default, so I just copied across the language file from the CMake bundle from Textmate and it worked perfectly.
It's definitely one to watch as development has been rapid.