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I use Ctrl + / to jump to the command bar (same as the search box, but with a > prefixed in the text). I then type "of" (for "open file") and start typing the name of the file to open. Possibly matches are automatically displayed by VS.
I also found that when using ReSharper, the familiar Eclipse shortcut for Open Type (Ctrl + Shift + T) is mapped to ReSharper's GotoFile command.
Unfortunately, VS doesn't offer this search-as-you-type functionality (and lots more).