Hi sblacklow,
Using Analyze > Fit Model you can specify your multi factor ANOVA manually or using the Macro > Full Factorial after selecting your factors in the column list. Here is a PDF showing the steps for fitting a 2-way factorial ANOVA (which generalizes to the higher-way designs):
http://www.jmp.com/academic/pdf/learning/04_two_way_factorial_anova.pdf
The steps for a 3 or higher-way ANOVA are identical -- when you use the "Macro" for "Full Factorial" JMP will populate the Model Effects section with the main effects, 2-way interactions, the 3-way interactions, etc. Be sure that your predictor variables are all marked as Nominal.
I hope this helps!
Julian