As far as we know these rings are just collected for fun. You get them naturally as you play through the game - Well, most of them at least.
Here’s a table telling how to obtain each.
||How to Get
||Spring Thanksgiving Festival - If your boyfriend/husband has orange heart or higher for you.
||One on the Starry Night Festival.
||One on your birthday.
||One after you get married.
||10th anniversary present.
||One in the mail (Itís some random lottery trickÖ)
||Take 100,000,000 steps and have it recorded on the pedometer (just keep the pedometer with you and you’ll get it eventually).