Organisation and Architecture of Quantum Computers
What you describe as current computers is known as the von Neumann architecture. This approach is one of many ways to think about classical computation and there are other classical approaches that might or might-not have relevant generalizations to quantum computing. The von Neumann architecture seems to be unlikely to be relevant to quantum computing, due to its difficulty from both the theoretical and implementation side.

However, as I mentioned on cstheory there is an article on implementing a quantum von Neumann architecture. They do this via superconducting qubits, of course the implementation is very small, with only 7 quantum parts: two superconducting qubits, a quantum bus, two quantum memories, and two zeroing registers. This allows their quantum CPU to perform one-, two-, and three-qubit gates on qubits, and the memory allows (data) qubits to be written, read out, and zeroed. Implementing a quantum superposition of gates is very difficult, and so the program is stored classically.

More likely models of quantum computing to be implemented include: measurement-based, topological, and adiabatic models. Typical implementations of these models look more like physics experiments (which they are!) than computers. Some of the common strategies for implementation include trapped ions, quantum optics, and superconducting circuits. 

The circuit approach has been placed on chips and in fact D-Wave (a spin-off from UBC in Vancouver) claims to have built quantum-like computers using the adiabatic model to implement quantum simulated annealing. They have managed to sell this computer to Lockheed Martin but their approach has been met with heavy skepticism.

Lastly, the NMR approach mentioned by @RanG. is interesting, but suspected to be not equivalent to full quantum-computing. It is equivalent to the one-clean qubit model (also known as DQC1) and suspected to be strictly weaker than full quantum computing.