I believe you would need a loglinear model to assess relationships between all three of your variables at the same time. Essentially, what you have here are counts in a crosstab: You have N1 cases of inanimate/nonpronoun subjects with animate/pronoun recipients and inanimate/nonpronoun themes, N2 cases of inanimate/nonpronoun subjects with animate/pronoun recipients and animate/pronoun themes, and so on for all possible combinations of your variables and their levels. What you are interested in is actually whether the frequency of observations across different categories is independent. Independence basically means that knowing what the value of one variable (say, agent) is will not help you predict the value of another variable (say, theme). This can be intuitively seen if you imagine an extreme case, say, animate agents only ever occur with animate recipients and animate themes. In this case, there is no independence, as knowing that your agent is animate immediately lets you know what value the other two variables will assume.
You would run your loglinear model with all three of your variables with their highest order interactions (agenttheme, agentrecipient, recipienttheme, agentrecipient*theme). The results will tell you if, for instance, the odds of having an animate theme is higher for animate agents than for inanimate ones.
