Well, yes, the primary function of the heart is to circulate blood; by replacing it with a positive displacement pump, you should be able to almost completely replace the heart's known physiological functions.
There are a few issues that I can see.
The first is the issue of clotting. You would likely need a steady supply of Heparin or other equivalent anti-clotting agent in order to prevent the blood from clotting within the mechanical parts (for all intents and purposes, blood can be thought of as magical and being able to tell when it's in something other than a vein or the body - so it clots... hell - sometimes it clots in the legs or brain when it's not supposed to).
The second is that vein and artery grafts are tricky as hell. Blood vessels of all types will grow and meander. The circulatory system of our body actually changes in response to things like exercise - vessels strengthen and weaken, branch and divide - it's very troublesome when you have chronic patients (such as in Dialysis) and things drift around. Likewise - the implant can see issues with veins it is connected to being unruly.
In the long term - it's questionable as to what the absence of a pulse would do. It may sound strange - but the beat of the heart produces a sudden rise and fall in pressure throughout much of the circulatory system. To believe that our bodies have not developed systems that function using that principle would be somewhat naive. I haven't been able to do a thorough review of the animal trials - but for a long-term "bionic" replacement, it may be better to use a pump that can emulate the throb - such as a set of diaphragm/piston pumps or a peristaltic pump (which, in theory at least, could use a grafted real vein and a set of rollers to prevent the blood from needing to come into contact with anything other than body tissue).
Otherwise - in Dialysis, we do something very similar. We take blood outside of the body, circulate it through a set of fibers, and then return it to the body. Flow rates are fairly slow (highest most prescriptions call for is 600 ml/min) but determined by both patient health and what can be practically drawn through the needles used as an interface. There are also limits on what can be done in terms of what is good for setting up an access (since this is all done outside of the body).
By no fluke, the best way of getting blood outside of the body is to access the arteries that are often contained beneath fat and muscle layers. Veins are easy to get to, but they have much lower flow rates. The popular solution is a fistula - a grafting of a vein and an artery together near the surface of the skin. There's a big push in Dialysis to move everyone to them who can tolerate the procedure. It has lower incidence of infection - but does require exercise to keep active and can end up being shut off by the body.
There's also some research that suggests there could be some consequences for the heart on the long term (because you are basically creating a short loop that allows blood to flow rapidly from the heart and back to it) - but considering your options are the mess of catheters, implanted grafts (that must eventually be re-implanted), and fistulas... or dying...
Anyway - since we're talking about replacing the heart, here, that's all of somewhat limited relevance.
