1. chose a record with noticeable surface dirt and noise (one that is replaceable!)
2. filled the laundry tub with warm water (don't use hot - it will buckle the LP)
3. added 3 squirts of dishwashing liquid (the biodegradable, gentle on hands type)
4. made a cradle for the LP out of wire
5. put the LP onto the wire (through the centre hole)
6. rotated the LP several times while washing with a micro-fibre cloth
7. rinsed off with cold water
8. removed the record from the wire and put on a lint-free towel on the table
9. gently patted the label with a paper towel to dry
10. left it supported on a V shaped piece of cardboard to dry (this only took 10 mins)
result: the visible dirt (dull areas etc) was gone. When played there was still a little surface noise, but the pops and clicks were all gone.
Update: I have now done my first batch of 15 dirty records. The process above worked fine, the only difference is having enough towels laid out for this number of records, and since I was doing a lot I skipped the part where the LP sat on a card - I just left them on the towel (don't forget to turn them over so both sides get completely dry). I have two copies of some titles so in some cases I washed one only to compare. As stated above, washing cleans the record. On some there is still noticeable surface noise, but generally there is an improvement. What washing cannot do is fix damaged grooves (whether caused by incorrect playing or rough handling), and scratch noises appear worse because the rest of the track is quieter.