Dropbox should work fine - just make sure all the files are uploaded in the same folders, in the public folder, and it should work OK. Link us to your game on Dropbox if it's not working and maybe we can see what's gone wrong.
Another new alternative is you can try publishing to the Chrome web store - if it's under 10mb you can make a Packaged App which is hosted on the Chrome Web Store itself.
VampyricalCurse, can you also link to the uploaded project so we can take a look?
Here are some tips to make sure it's working:
1) Press Ctrl+Shift+J in most browsers (F12 in IE) and check the Console for any error messages. You might get something like "GET soundeffect.ogg failed", which means the file soundeffect.ogg is probably missing.
2) Make sure your server is set up to serve the following MIME types:
.ogg - audio/ogg
.m4a - audio/mp4
.appcache - text/cache-manifest
By default servers do not serve files with extensions they don't recognise. So if your server is not set up right, you'll get 404 Not Found even if you directly visit the URL of one of the .ogg or .m4a files you uploaded. (Visiting the URL directly is actually a good way to test this.) This would explain why no audio plays after uploading.
Please always link us to the project you've uploaded if you're having trouble though - we can poke around and test these things ourselves, but we're stuck guessing if you don't give us the URL.