It is possible - with the newest software update (version Version 3.0.0-6U), you can transfer software, sounds, and images from one 3DS to another using the System Transfer found in the System Settings menu. However, if you no longer have access to the Nintendo 3DS, the transfer cannot be completed. It is all transferred using local wireless communication, which means you need both your old 3DS and your new one to do this transfer.
From Nintendo's support page:
For the most part, information on the target system (system being transferred to) will be overwritten by the information on the source system (system being transferred from). See below for specific details:
- All content downloaded via the eShop on both systems (DSiWare, Ambassador games and status, 3DS Downloadable software, etc) will be combined together onto the target system.
After the transfer is complete, titles will be available to re-download from the Nintendo eShop.
- Any game data that is duplicated on both systems is overwritten by the data on the source system.
- The eShop balance from both systems is combined together. (Total amount cannot exceed $200 or the transfer will be canceled.)
- Data associated with any pre-installed software or applications (Face Raiders, Streetpass Mii Plaza, etc). Only the data associated with the source system is retained.
- Photos, Videos, and Sound files. Any photo and sound files that exist on the target system’s internal memory will be overwritten. However, photo and sound files saved to the target system’s SD Card are still retained.
- System Settings from the source system (Parental Controls, Internet settings, etc)
- Save data associated to DSiWare and Nintendo 3DS Downloads
- The WFC ID associated to DS online games.