Apple has released a statement on the matter, saying it's aware of the problem and will have a fix "in an upcoming software update".
Apple has reportedly seeded an iOS 7.0.3 update to developers for final testing, which could be ready to download next week. Let's hope this is the fix Apple is referring to.
Although it seems a lot of people are experiencing the problem right now, Apple says it's only affecting "a fraction of a percent" of users.
For now, one workaround we're hearing of is to turn off iMessage, hit Reset Network Settings, then turn iMessage back on again.