I have solved it myself:
<setter property="Background" value="{Binding RelativeSource={RelativeSource Self}, Path=Content.Text, Converter={StaticResource ResourceKey=AlarmConverter}}" />
... and the object passed to the converter is the string in the cell. And that is what I needed.
[Why can't I delete my solution - and can't I add my own solution properly right from the start, not only after an other is posted?]