What is the easiest way to re-use test cases across multiple test cycles

You might find the Filter feature useful in this scenario.  It enables you to view test cases by criteria so as long as you 'tag' your test cases that you want to re-use, e.g. the Component, Labels or a custom field, you can then filter to those specific tests and add them into the cycle(s) where they're needed.

You can find the filter feature in the Test Cases area (1) and in the top right-hand corner of the window where you add test cases to your cycle (2).  If you are going to use this feature then you will probably want to save the filter for future use.  It's a simple but effective tool.

Hope that helps but let me know if it doesn't or if I need to explain further?

1: Test cases area

Choosing criteria:

Or choosing a saved filter (to save a filter the button is on the far right of the filter bar

2: filter button in test cycle > test cases > add test cases