Cord-like Feeling Running Along Crease After Breast Augmentation?

If I understand you correctly you have a cord running paralled and NOT perpendicular to your breast fold.. This is either your incompletely release inframammary fpld or the remnants of your pectoralis muscle cuff along the costal margin subsequent to release of the costal margin origins ofthe pectoraliz major muscle 

It is most likely just internal scarring that should subside within about a month. It is very common. However, if it is something that concerns you, it is never a bad idea to have it checked out by your surgeon just to make sure.
Best regards,
Dr. Speron

What you are describing is not uncommon at all. I would discuss it with your surgeon next time you see him so that he can examine you just to be sure since we can't examine you over the internet. However it's not uncommon to feel lumps and bands under the skin while you are healing. Most of these things will go away over the next several months, and sometimes may even take up to a year.

Bands such as yours are often inflammation that will resolve.  Normal inflammation associated with healing actually is most prevalent at about 6 weeks postoperatively, and with what is called remodeling of the tissue, your problem will likely resolve in the next few months.  

I am impressed you know of Mondor's Disease!
If your incision is under the breast crease, you most likely are feeling scar tissue under the area of incision. Remember, there is constant remodeling of collagen under your incision for 6-12 months after surgery.

What you are feeling are not uncommon. These are scars internally that are forming and in all likelihood will disappear over the next 4 weeks. Very often I will release the Mondor bands with pressure. The horizontal band could be the inframammary crease if the breast was lowered or you had a tuberous breast.
I would go back to your plastic surgeon and discuss this with him or her. It is probably normal healing but instead of worrying why not go to your doctor and know once and for all.