Well, your Sony S370's about as close as it gets without buying a dedicated media player. The only difference between the Sony 370 and its wireless media playing counterpart is, er, wireless capability.
Spend £40 on the likes of Belkin's Wireless N Router, plug it in the back of the Sony's ethernet port (thereby saving on the £70 you'd need to spend on the Sony dongle and avoiding the proprietary dongle issues) and you're done. It's got a USB capability on the front, so you're able to use that for memory sticks.
I was thinking about getting the Sony media player, but have now ditched said plan. Will probably read up on wireless routers, connect it up to our main house router (so it acts as an extender and avoids the need to send wires across the hall, etc. etc.) and do it that way.