A. Switch temporarily to a WordPress default theme (Twenty Eleven etc.) and see if the problem goes away.
If switching to default theme doesn't help it could be one of the plugins you are using that is causing the problem.
B. Disable (temporarily) all plugins other than Contact Form 7
If doing that solves the problem, you can then add back each item - one by one, until you find the possible cause of the problem.
Note: You may need to clear the browser cache between each one.
Using on_sent_ok in Additional Settings
Using on_sent_ok: in Additional Settings is often mentioned in the CF7 Support forum as a frequent contributor to problems.
You can temporarily clear out any items you have in Additional Settings to see if this is causing your problem.
Use Firebug or Chrome Dev Tools
- Lack of errors and errors showing a symptom rather than the cause is unfortunately more commonly the case than not.
Remember that a problem in one script can cause issues in another.
If you identify a conflict with a particular plugin or theme, you can & should write to the plugin or theme developer and suggest they might have a possible conflict with CF7.
CF7 is one of the most popular plugins in the WordPress Plugin Repository, so it makes sense for developers to deal with any potential conflicts.
Also if a plugin or theme conflicts with CF7, it probably also conflicts with many other plugins. It can be a sign of a poorly coded theme or plugin that people should avoid using.
When Plugins Collide: Best Practices For Dealing With WordPress Plugin Conflicts
Conflict Diagnosis Guide for WordPress plugins
Still have questions?
If you find you need more detailed advice, after reading and working through the above links, please create a new question here on WP Answers explaining your specific problem in detail.
1. A link to your Contact Form 7 form
2. Detailed description of your issue.
This should allow others to understand your actual problem and offer a possible solution.
Read How to Get Better, Quicker Support for detailed guidance on asking questions.