In the "mainstream" market there is e.g. Norwegian Cruise Line operating a concept called "Freestyle Cruising". There are formal nights on this line's ships, but you needn't take part in them. This means that of several restaurants onboard (operated on an open-seating basis) only one will host the formal night dinner while others are casual throughout. There are usually at least two main restaurants per ship, with their menus being virtually identical, you you don't miss anything food-wise.
A more upscale alternative is Oceania Cruises, which has no strict dress code at all. And while many Oceania passengers enjoy to dress up, it is not compulsory.