The only phrase I know for this is riding pillion.
This phrase immediately came to mind when I saw the question, but it's sufficiently rare that I checked I had it right.
I've never heard either of the expressions suggested in other answers.
It may be that pillion is unfamiliar in the US. The GloWbE corpus shows only 13 instances of "ride pillion", but 5 of them are from the UK, and none from US or Canada.
Edit: looking for just "pillion" in the GloWbE corpus, I find 453 instances, of which only 2 are from the US and 3 from Canada. There are 86 from the UK, but the highest number is 107 from Pakistan.