Today, Apple made a statement to The Wall Street Journal acknowledging an issue with iMessage delivery failure in iOS 7. This problem seems widespread based on the number of users reporting it on Facebook and Twitter, but Apple says otherwise and is working on a fix for what they believe to be a “fraction of a percent of iMessage users.”
“We are aware of an issue that affects a fraction of a percent of our iMessage users, and we will have a fix available in an upcoming software update. In the meantime, we encourage any users having problems to reference our troubleshooting documents or contact AppleCare to help resolve their issue. We apologize for any inconvenience this causes impacted users.”
With the software update on the way, users will have to stick to the process of rebooting for a fix. But, if a reboot isn’t satisfactory you can go an alternate route through Settings by disabling iMessages, reseting your network settings, and re-enabling iMessages, a successful fix (for the most part) that originated from user trial and error.