We used Rainier, who offered us a "slow boat" option. Our stuff went into three crates, and when enough other crates were leaving our area for the UK they were all bundled into on cargo container. They picked up our stuff around the third weekend in April and we had it by early July, which was what we wanted - we came over in early May not knowing where we'd be living and wanted the time to house-hunt.
The option we chose gave us no weight restrictions - which was good, as I had close to five dozen cases of hardback books and 400+ LP record albums. Basically, if it fit in the crate, regardless of overall weight, it went for the same price. They managed the move, contracting with a local mover to crate up everything on the sending end. They were good about the estimate beforehand (get at least three estimates - have them come actually look at your stuff, don't use a quote off the internet or phone). They arranged for Britannia movers here in the UK (a bit lackluster, but adequate) to uncrate our stuff at their warehouse and bring it on a truck up to us in Scotland and into our flat. Throughout the trip we had the boat's name and could track it on a GPS map tracker that Rainier provided to us.
Pretty much everything went as it should and I can recommend them to you.