Here's a simple three-step fix that seems to be working for many of those affected.

While we wait for that software update we still want to send messages so try this fix.
ios 7 Settings icon
Messages ios 7 fix
Go to the "Settings" (via the grey cogs icon on home screen), and scroll down to "Messages" (green speech bubble icon) in the column that starts with iCloud.
Messages ios 7 switch off
Tap that and slide the top bar to turn off "iMessage".
ios 7 general settings imessage fix
Then go back to Settings and scroll to "General". Tap that, and scroll all the way to the bottom of General for the option "Reset".
ios 7 reset option
Tap Reset, and tap the option "Reset Network Settings". A further option appears in red at the bottom of the screen saying "Reset Network Settings".
ios 7 reset option
ios 7 reset network settings
Tap that, and your iOS 7 device will go into a quick reset. Please note: resetting network settings will make your device forget Wi-Fi passwords, so you'll need to add these back in. Make a note of your favourite network passwords before this step. Most people have a only a few passwords they use regularly so inputting them again is a small price for getting your text messages back, but if the thought horrifies you, please take heed of this warning.
Once this is done, go back to "Settings" and choose "Messages" again. Slide the top switch again so that '"iMessage" is switched on again.
That should fix the problem of iMessages not working in iOS 7. As some of the comments below this article point out this simple fix doesn't always work and we look forward to Apple getting round to a software update that eliminates the bug once and for all.