For most healthy people, caffeine could cause a short, temporary increase in blood pressure. If you don't have high blood pressure you can still feel your heart race after your morning cup. Avid coffee drinkers can build up a tolerance to the caffeine and may not experience effects after a cup. So is it true that coffee is bad for the heart?
If someone already have a heart condition, probably she should ask such questions to her physician.
As it's noted in the question itself, it's a known fact that caffeine may increase blood pressure temporarily. If someone is on the edge of an hearth attack, coffee may trigger this (by increasing the blood pressure). As well as, running after your grandson or arguing hot on some topic could trigger, too.
In summary, under normal circumstances, a temporary blood pressure increase should not be a problem for your health if you're a healthy person. However, we are not M.D.'s here, so, please go ask a physician first if you have doubts. Drink decaf if you feel safer.