I've been having issues with iMessage since upgrading to 7.0.2, it looks like they're sending and then it just stalls and does nothing, no error just no send. Tried restarting a few times and no joy, a google search brought me here and I tried a few things people suggested and my phone was still stalling on send.
Until, I disabled background app refresh, it sends with no issue now. Not sure how the two are related but it worked for me!
If you're having these issues, it's worth a go because it's incredibly annoying.