What is the best way to integrate Tsheets with a popular calendar software/organizational tool?

On a computer, go to Schedule. At the top left of the Schedule, click Actions > Subscribe. In the Subscribe to Calendar window, select the schedule to which you want to subscribe. Click Copy Link > Close. If no Copy Link button is displayed, click Ctrl+c (for Windows) or cmd+c (for Mac) to copy the link. Go to your non-TSheets calendar, locate the Subscribe feature, and paste the TSheets link into the appropriate field. Below are instructions for the current versions of the most commonly used calendars.

There is no native way to integrate with another calendar but there is an API you can use to develop such an integration. New integrations are added all the time as they become popular requests.