That just indicates that the module didn't automagically load on boot. If you load it manually, you should get a wireless interface ra0:
So, I rebooted and ran lsmod from a blank start.
When I did so, it listed nothing at all as having been loaded.
As well, you should then get some activity in the kernel messages:
sudo modprobe ra0
dmesg | grep -e rt2 -e rt3
I don't know which one you downloaded. The one I referred to in my example, DPO_RT3562_3592_3062_LinuxSTA_V188.8.131.52_20101217, is the latest available on Ralink's website. It is pretty old in Linux terms, almost a year! It produced many warnings, but no errors, when I compiled it on my own system. It is clearly intended to be correct for your device:
p.s. is it possible that I've downloaded the incorrect module?
This is a tough issue. The intent is to fold all the 'staging' modules into the mainline module in kernel >= 3.0. Your device is intended to be driven by rt2800pci. Before you modprobe rt3562sta, you might try rt2800pci:
chili@LAPTOP60:~$ modinfo Desktop/Forum/L_/DPO_RT3562_3592_3062_LinuxSTA_V184.108.40.206_20101217/os/linux/rt3562sta.ko | grep 3062
Do you have a wireless interface wlan0? Does it try to connect? Any clues here?
sudo modprobe rt2800pci