I am a US Citizen and have a 10-hour layover in Heathrow - with checked baggage. I want to go into the city, leaving checked luggage. What obstacles might I encounter?
Assuming that your 10-hour layover is not overnight, and that your arriving and departing flights are booked together, etc., you can generally expect to have your baggage checked through such that you won't see it at all in Heathrow.
As a US citizen (traveling with a US passport) you're free to leave the airport and tourist around while you wait without any further formalities. However, beware that you'll generally need to do this as soon as you disembark from your arriving flight, since you then have a choice between following the signs for "arrivals" (immigration) or "connecting flights" (transfer security). If you end up going through transfer security, you end up in the secured departures area, where there's no routine route out to landside.
You will need to wait in line for immigration and answer a few questions about what your plans are -- if you answer truthfully and are ready to back that up by showing your onward boarding card if challenged, the risk of any problems there is practically non-existent unless you're already in trouble with the UK for some reason.
Just be sure to be back at the airport in time to get through security and be at your departure gate in time for boarding.