If you're effective at getting employees to take more initiative, you have to provide them greater autonomy, to help them succeed. All employees need to have a say in how they do their work to make it more meaningful. After you enlist your employees to make suggestions and improvements, you need to encourage them to run with their ideas, take responsibility, and champion those ideas to completion.
Allow employees to approach anyone they need for help, give them the authority to make decisions or use resources, and permit them to take the actions that are necessary to get the work done.
In research on employee preferences at work, "autonomy and authority" and "flexibility of working hours" were two of the top motivators for today's employees. To the extent that you, the manager, or the organization is able to provide those motivators for employees, their morale and performance will be positively — and significantly — impacted, and they'll do their best work possible.
No one likes to be micromanaged. The vast majority of employees would prefer to determine how they work best. In other words, they'd prefer to be assigned a task and allowed the freedom to develop a work plan that suits them. As a manager then, focus on the end result and allow your employees to put their own imprint on the job — to decide how best to achieve the result. Take it a step further by allowing employees to pick and choose the projects and responsibilities they can work on as a reward for having previously done a great job.
Here is where truly knowing your employees becomes important: Understanding their strengths and weaknesses allows you to properly assign projects and tasks, making suggestions for assignments that you know the employee would likely value and helping that employee develop new skills in the process.