I have a date in C3 (October 12, 2009) and would like cell d3 to
display that same day, but 3 years later (October 12, 2012). Is there
a formula that I can use to make that happen?

One way:
=DATE(YEAR(C3)+3,MONTH(C3),DAY(C3))

Notice that for 02-29-08 (leap year) the result is 03-01-11.