I experienced this. The palpitations were so bad that they would keep me up all night. In my case, it turned out that I was quite anemic. The problem went away with iron supplementation.
Not to alarm but I should mention that heart palpitations/chest constriction should always be reported to your doctor. I mention this because of what I've recently gone through and am still going through. I am hooked up to a Holter monitor as I type, having just gone through a chest CT and echo too. I was told throughout all this that pregnancy raises the risk of pulmonary embolism and pneumonia, both of which need to be diagnosed and treated. Pulmonary embolism is life-threatening. Also, there is another condition called PPCM that strikes in the third trimester, but when I first mentioned my palpitations to my OB (in the first tri) she still sent me for an EKG nonetheless, to rule it out. The odds that it's any of these conditions are very low, but just wanted to make sure it was mentioned to you.
It is more than likely anemia or thyroid, as others mentioned, but I would really recommend discussing it with your doctor to rule out something more serious like the conditions above x