Various Famous Cities, Present Day
Amy and Dan's trip takes them all over the world. Their grandmother, Grace, whose idea it was to send everyone on this wild clue hunt, was an explorer. The race around the world almost seems designed so everyone can follow in Grace's footsteps and learn a few secrets about well-known cities like Philadelphia and Paris along the way.
Although a year is never given, we can assume The 39 Clues takes place in the vaguely defined "present." Amy and Dan have access to modern conveniences like cell phones and the Internet, but they're not zipping around in flying cars or teleporting or anything sci-fi like that.
And if you pay attention, you'll soon realize that the focus of The 39 Clues is the past. Amy and Dan have to look into history to find the right trail. Only by seeing where Ben Franklin has been can Amy and Dan figure out where they're going.
Other than that, the settings fade to the background. They serve symbolic purposes, like the mansion (see "Symbols, Imagery, Allegory"), and act as backdrops for the crazy action that goes down.