I ride in toe clips - no straps. Others have said they do, too. There was a thread I started on this wit the same question.
My feet stay in there fine. I don't pull up hard on the backstroke, though. I do light spinning and some light mashing.
Keeps my feet on the pedals on bumpy raods and descents.
I use MKS Touring and MKS Deep Toe clips, M size.

Watch out with the GR-9's. They are narrow. They weren't wide enough for my 9.5 shoe size. My foot would feel like it was rolling off the outside edges of the pedals. So I got the Touring pedals. Plenty of support across bottom of feet now.