Should I link the timesheet to our project management system?
Only if doing so will give you back a good return on investment. If you use a project management system like Microsoft Project, Primavera or Open Plan you may find that doing good planning and comparing that plan with the actual results makes your organization better at managing critical projects, but this type of discipline isn't for everyone. Some projects are so simple that they don't warrant a huge project schedule. Others are so out of character for the organization that they would be better managed outside of the corporate systems.
But, for many organizations committed to project management as a key corporate practice, integrating the project schedule and timesheet can return great dividends. To know if your organization is ready, one key question to ask yourself is this: "Is our project planning system and culture mature enough to integrate with a centralized timesheet system that is used by all staff?" If the answer is no, then put off integration until you're ready.