We've only used Regent's air once when we booked a cruise at the last minute. From Boston to Anchorage we flew Northwest/Alaska Airlines and returning from Tokyo we flew American.
All flights were in a fare class that meant we could not get frequent flyer miles. That isn't always the case, but sometimes. I don't think you'd know until you get your airline info and look at the fare bucket. If it's heavily discount, you won't get the miles.
Of course, always attached your frequent flyer number to all tickets. Mistakes do happen and you could end up getting credit even though you technically shouldn't.
Andrea