I had the same situation. What I did was to anchor 1x3 strips to the floor using a power hammer. I laid them 16" on center on one axis and 24" on the other axis. I then placed 1" hard foam insulation between the strips and covered everything with plastic taking care to duct tape the seems. I then nailed down 3/4" plywood and carpeted over using a good quality pad under the carpet. Six years later I’m still happy with the results.

That I can definitely do!!

OK, dumb question: Can a power hammer hammer right into concrete?

Danger!!!

A power hammer can be a dangerous tool to use if you don't know what you're doing. It involves gunpowder and special hardened nails. Given your question; I strongly suggest you call in a professional for this phase of the project.

This link will explain what a power hammer is and its' uses.

http://doitbest.com/shop/product.asp?mscssid=N32FC86MWMS92PNU00JP49NKVSH0F7LD&dept%5Fid=1946&sku=355350

Be sure that any wood placed in direct contact with cement is treated. Untreated lumber tends to draw moisture from the cement and will eventually rot and can also cause mould problems.

Alternative method, Hammer drill and nails. They now have a special nail, that once driven into concrete, cannot be removed. Wish I had those on my last project, used screws and the darn things stipped too easily. I ran 2x4's flat on the concrete and tried a high velocity nailer on it, but the concrete was so hard tha the nails bent. A low velocity just managed to chip the floor. I wound up using a Bosch hammer drill and screwed the boards down to the concrete 16" OC and put 1 1/4" foam panels between them, then screwed and glued 3/4" OSB down to the 2x4's. I have a nice warm solid floor now in what used to be a garage.