Your follow up habits and skills are responsible for 80% of your sales. — Jeffrey Gittomer's Sales Bible

This is true in sales, and it's also true in job hunting.

When you first connect with someone, they might be on their phone. It might be easy to just click "accept" without replying. It doesn't mean they don't like you. It just means they are semi-mindlessly doing stuff on their phone, just like most of us do a lot of the time. Turn this into an opportunity. Follow up.

When you send your resume to HR and don't hear back, it doesn't mean you were rejected. It means, like us, the HR person is overwhelmed and hasn't gotten back to a pile of emails. It's not personal. Turn this into an opportunity. Follow up.

Following up is everything.

Here are the three most important scenarios to follow up

1. Someone accepts your LinkedIn connection but doesn't say anything. Follow up.
2. HR manager tells you to send your CV. Doesn't reply within 1 week. Follow up. (In general, unless you and a person explicitly talked about a specific timeline, e.g. in two weeks, never let more than one week pass with no contact. Out of touch, out of mind. Follow up.)
3. After any call or interview with anyone. Follow up to 'solidify' your enthusiasm towards this person.

For scripts, see: LinkedIn Connect Scripts / Templates