There are many, many problems with building a good artificial heart:
Turns out the body doesn't respond well to typical mechanical pumps - the pumping action of the heart is quite specific, and body circulation suffers with many common "pump" designs. The most successful designs emulate the heart by building, essentially, heart shaped bags and valves and using hydraulic fluid to inflated and deflate them to move the blood around just like the heart does.
The body also hates foreign materials. Not only do they have problems with rejection (ie, immune system overreaction to plastics, etc) but there are only a few biologically compatible materials that you can get good, long term connections between human tissue and artificial materials.
Further, even in a good connection there may still be places where blood flow is not smooth, which collects and/or creates clotted blood, which can cause stroke or blood clots in other parts of the body when these small clots break loose. The early artificial hearts had this problem to a significant degree.
Assuming we fix all the above problems, there's still the problem with a power supply. The most successful, long-ish term heart implant right now requires connection to an external power unit. Those connections must go through the skin, and provide a site for infection. That one is meant only to act as a bridge to transplant, and is currently FDA approved only for hospital use - in other words you can't go home or do much with this artificial heart. They are doing a study to see if their portable power unit may be used away from the hospital. The wikipedia article talks about the fully implantable heart, but it is only FDA approved for those patients who are not eligible for a heart transplant, where their own heart is failing - in other words it extends the life of terminal patients, but it does not truly replace the heart.
So, while it's conceptually simple, the devil is in the details.
