Yesterday I registered a Russian visa for my friend from Germany at a Russian post office, so below is the first-hand step-by-step explanation of the whole process, which is quite new (fully working since this summer).
According to the new rules any Russian citizen with "propiska" (registration) can register a foreigner at his/her apartment at a local police station or any post office. The whole procedure takes about an hour (because of the forms you need to fill out and queues) and costs about €5.
Another alternative is to contact your travel agency and they will be able to do it for a fee (usually $20 to $50 US). If you're staying in a hotel, they'll usually do it for free.
If you don't have a registration, the border officials will not bother you anymore, as it's not their responsibility to check the registration -- they'd only need the migration card back. However, if you get stopped by police without registration you may be fined 2000 to 5000 Rubles ($75 to $200 US). They don't usually stop people often on the street (hasn't happened yet with my friend), but if you go to the main touristic spots the chances to be stopped for a routine check are quite high.
So here's what we did...