Ironically, one of the things that keeps many couples from praying together is the fact that it's so incredibly awkward. Different kind of awkward, but still, awkward. Fear not, I'm not about to try and talk you into it for the purposes mentioned above. However, I would just like to say that whatever the fringe benefits may or may not be, couples need to pray together. I tell our high school students all the time - look for the person who will pray with you and for you.
When my husband and I were engaged and living in separate states, we used to pray together over the phone. That was more awkward for my roommates than for me. As a married couple, we've had periods of time when we've been very faithful to pray together and periods where we've been awful about it. I can guarantee you the times we're praying together are our best times, our most genuine times and the times we've been the closest and gone deeper in our relationship.
I'm thankful for a husband who invites me to pray with him even on the nights when I'm annoyed with him and would rather just go to bed. I'm really thankful for a husband who invites me to pray with him on the nights when he is rightfully annoyed with me and doesn't really want to talk with me let alone pray with me. Mostly, I'm thankful for a Savior who invites both of us into His presence even when we're so unworthy to be there, but at least we're there together.
I know it's awkward, but today, ask your spouse or the person you're dating to pray with you. I promise you it will be awkward. I promise you it will be worth the awkwardness.