Welcome to Premiere. I also shoot on a 5D and it has been working great for me since CS6 came out.
You have a few options, and an easier workflow than with FCP, since Premiere works with H.246 natively without the need to transcode.
You can either just dump the footage to a HD, and import the .movs right into your bins in Premiere and start editing...or oven preview to footage right on the card from within Premiere and bring over just the footage you need. You have access to hoverscrub from the import, so you can shuttle through your clips and pick only the ones you need.
Another option is to use Prelude, which comes bundled with the suite. This program is designed for importing, transcoding, cataloging, organizing, and adding metadata to footage before you begin your edit....it is pretty powerful. You can use this program to copy over from your cards and also transcode or make backup copies at the same time. There are a lot of tutorials out there, but it is an awesome program.
The only caveat about editing H.264 files natively is that you have to have a capable machine, since it is pretty CPU intensive. If your machine can't keep up, you may want to go back to ProRes transcodes with your files.
do I just dump the footage from the card onto a HD and edit from there?
Yes, that. Just be sure to copy the entire card, and keep the folders intact.