I had the EXACT same thing happen to me during both of my pregnancies. I remembered friends saying to expect it when my belly was big and pressing on all of my organs, but at three months and not showing at all, I thought "What is going on??!!" The tightness in my chest and shortness of breath typically happened to me when I was walking to the train, although it happened at other random times as well. I wouldn't necessarily be walking too fast and often it happened when walking at a leisurely pace. I don't have asthma and never really suffered from allergies, so I chalked it up to pregnancy. My OB didn't have any good advice about why it was happening and didn't seem to think it was anything to be too concerned about, so I just tried to very consciously take slow, deep breaths when I'd feel it tightening up. That helped a bit until it passed, which would take about 5 minutes.
One thing that also happened to me while pregnant, and I'll share in case it does/is happening to you, is that I would get a woozy feeling in my head when sitting at my computer for a long time. I work from home and would go hours without getting up, sometimes with my legs crossed. This was something my doctor was concerned about because basically I wasn't getting enough oxygen to my brain, which was making me feel dizzy and like my equilibrium was off. The fix was simple: Just get up and walk around every once in a while. It really helped -- and even though no longer pregnant the wooziness still creeps up sometimes when I've been sitting too long. Amazing how the body tells us when we need to move around -- or slow things down.
Good luck. I'm sure you and your baby will be just fine. Despite my little issues while pregnant I ended up with delightful, healthy children. Kudos to you for knowing yourself so well and taking steps to make sure everything stays good for you and your little one.