There are several third party apps that can handle mp4. Check out oPlayer in the App Store.
iTunes does .mp4 just fine. Most podcasts are mp4.
After thev'e been imported into iTunes, make sure they show up as what you want; podcast, music video, etc. If they don't then right click on the file, choose Get Info, and change the Media Kind under the Options tab.
Make sure you are set to sync that type of Media, either all or the various slected options. Then sync. The file will show up in the Video's app. If it's a Music video it may show up in the iPod app too, but when you play it, it will use the Video app.