I tried a traditional money clip for a while, but it made it a hassle to keep cards organized, so I would really only recommend that if you mainly use cash.
Now I use a fairly simple, lightweight wallet/magnetic clip like this. It doesn't add much bulk, but has enough compartments to hold an ID, a few cards and some cash.
To each their own, but using a plastic clip to hold your money seems more like making a competition out of being minimalist than a practical solution.
I've been using this money clip for quite some time and it has been serving me well :)
It's not a bifold, but it does have quick access to the essentials. Only downside is sometimes the cards (I put multiple cards in each slot) get stuck together, I think it's trying to tell me to lose weight...