Note: This post applies only to males.
I always see male divers wearing bathing suits or board shorts under their wetsuits. This requires awkwardly stuffing all the excess fabric into the wetsuit legs, and not to mention, the suit tends to ride up. I did it once for my Open Water certification and hated it.
There are options, however. Even if you would never wear it otherwise, I highly recommend wearing some kind of form-fitting lycra swim material. This doesn't have to be the legendary Speedo briefs; I wear something similar to these Speedo jammers and think they're great. You can also get the square shorts for something smaller than the knee-length jammers, but with more coverage than briefs.
Once you switch it will be hard to go back!