Your flash cart will work on the 3DS, as long as you have the skills to hack your device and make it work.
The 3DS has only been out in Japan for a few days, but that's apparently long enough for flash cart manufacturers to hack the device and get the carts working. The cards apparently won't run 3DS games, but seems like Game Boy and DS games of all flavors, as well as homebrew content, work just fine.
There a number of videos showing various cards working; one shows Mario Kart 64 running from a flash cart, while another shows Link's Awakening running on emulation. According to one of the videos, the 3DS has a white list of DS games that will run on the device, which funnily enough doesn't include the flash carts. In order to get them to work, this white list had to be hacked, which means that it's not something that people can easily do at home.
This isn't the best of news for Nintendo, but it's something that it planned for as part of its anti-piracy efforts. Back in October, Nintendo said that as well as downloadable updates, it would be including firmware updates - which would reblock the cards - on 3DS games. It's also worth mentioning again that this isn't affecting 3DS games yet, and there's no way of telling when, or if, it will.