I had the same issue at first, but hadn't had to kill it. It just seems that the device need a while to refresh all information, like the covers and so on. If this takes too long ICS itself is shutting down applications, which it assume to hang. This actually can happen with all your applications.
As I had the widget on one of my main screens I saw the name of the last song but not its cover. I then started the song over the widget and the application reloaded the data. Afterwards I was able to open the music player as normal as the data got put into the internal cache.
If this doesn't work either check your music files with a file explorer or as sugested above try to restart your phone again (after an ICS update its usually shutdown and has to be started fresh anyway).
Sony should however optimize the process of loading covers and other meta data in the background for the Walkman App, to avoid having the main application got shut down if it takes a bit longer.
