Yep. Just log into the Disney website, click on "My Reservations" and they should be listed there.
If they aren't (typically because they were made before the new system was launched), then you'll have to retrieve them by confirmation number (an option on the right of the page). For this you will need the Dining Reservation confirmation number as well as the phone number that you entered.
If you do not have the confirmation numbers or the phone number, or the site is giving you issues (happens often enough), then you'll have to call.