SketchUp uses the [supported] operating systems standard browser (IE/Safari) in order to keep the installer file small and to have a " "consistent" " platform for webdialogs.
The chance is quite low that SketchUp would work with any random browser or to make it work with a linux browser. This is not due to linux, but just because Wine mimics Windows as much as possible so that Windows applications work best together with Windows versions of other apps (SketchUp with Google Earth, Building Maker with Firefox.exe). So the only way to go would be to integrate Chrome into SketchUp.
Although I back your request for a native Linux version (or at least a completely packaged version with wine in the Ubuntu Software Center), the best you can do for now is make webdialogs work in Wine by following this tutorial:
1. Go to ~/.wine/drive_c/Program Files and rename the folder 'Internet Explorer' into 'Internet Explorer backup' or something.
3. Then select the checkboxes for 'gecko', 'ie8' and 'flash' (for streetview imagery).
4. After installation, delete the newly created folder 'Internet Explorer' and rename the backup into 'Internet Explorer'. This sounds stupid, but it works!
5. Open Applications ⇒ Wine ⇒ Configure Wine. On the "library" tab, add an overwrite for wininet (this makes uploading work) and for ctl3d32 (this can be helpful for the component options/attributes window). Those two should be listed as "(native, built-in)". Also make sure you don't use Windows msstyle themes.
I hope this helps!